1. Greeting and Shaking Hands after Prayer: 
Abu Hurairah (radyAllaahu ‘anhu) reported that the Prophet (sallAllaahu 'alayhi wa sallam) said: "If one of you meets his brother, then he should greet him with Salaam. Then if a tree or a wallor a rock should come between the two of them and then he meets him again, he should greet him with Salaam again." 
In this hadeeth, the Prophet (sallAllaahu 'alayhi wa sallam) has commanded the Muslims, when any of them encounters his Muslim brother to greet him with Salaam. This is because of what the greeting consists of from the growth of unity, eradication of hatred and the bringing about of love. The command in this hadeeth is only a recommendation, meaning it is strongly advised and encouraged, and not obligatory.
And there is no distinction in this matter between one that is in a masjid or outside of it. Rather,the authentic Sunnah indicates the prescription of greeting those who are present in the masjid with Salaam, regardless if they are in prayer or not.
Ibn 'Umar (radyAllaahu ‘anhu) reported: "That the Messenger of Allaah (sallAllaahu 'alayhi wa sallam) went to Qubaa (masjid) to pray and the Ansaar came to him while he was in prayer and greeted him with Salaam. So I said to Bilaal (radyAllaahu ‘anhu): 'How did you see the Prophet (sallAllaahu 'alayhi wa sallam) respond to their greeting, when he was greeted with Salaam whilst praying?' He (Bilaal) said: 'He said like this' -and he (radyAllaahu ‘anhu) stretched out his palm. And Ja'far Ibn 'Awn stretched out his palm and put the inner part of his palm facing the bottom and the backside of it facing the top." 
Imaam Ahmad and Ishaaq Ibn Raahawaih both formulated their opinions according to this hadeeth. Thus, Al-Marwazee (rahimahullaah) said: "I said (to Ahmad): 'Can one greet the people with Salaam while they are praying?' he said: 'Yes' and then mentioned the story of Ibn'Umar when he asked Bilaal. So then I asked him: 'How did he respond?' He said: 'He used to gesture with his hand.'' Ishaaq said: 'It is as he said it is.'"  This is the opinion that Ibn Al-'Arabee preferred, where he said: "The gesturing of the hand during prayer can be done for responding to the greeting of Salaam and for a matter that occurs during the prayer. It can be also be done due to a necessity which calls away the attention of the one who is praying. If it is done in order to respond to a greeting, then there are authentic reports concerning that, such as the Prophet's action in Qubaa." 
The proof for the legislation of giving Salaam after finishing the prayer, in the masjid, is the famous hadeeth of the man who prayed incorrectly. Abu Hurairah (radyAllaahu ‘anhu) reported that: "The Messenger of Allaah (sallAllaahu 'alayhi wa sallam) entered the masjid and a man entered after him. He prayed and then approached Allaah's Messenger (sallAllaahu 'alayhi wa sallam) and greeted him with Salaam. So the Messenger of Allaah (sallAllaahu 'alayhi wa sallam) returned his greeting and said: 'Go back and pray, for indeed you have not prayed. 'So the man went back and prayed just as he had prayed before, and then went to the Prophet (sallAllaahu 'alayhi wa sallam) and greeted him with Salaam - and he did this three times altogether." 
Shaikh Al-Albaanee (ha) said: "Sideeq Hasan Khaan uses this hadeeth as evidence for: 'If one should greet his brother with Salaam and then encounter him a short while later, then it is recommended for him to greet him with Salaam second and a third time.' " 
And Al-Albaanee (ha) also said: "There is evidence in this hadeeth for the prescription of greeting those who are in the masjid with Salaam. What indicates this is the hadeeth in which the Ansaar greeted the Prophet (sallAllaahu 'alayhi wa sallam) with Salaam in Masjid Qubaa. In spite of all this, we can still find some fanatics that don't attach any importance to this Sunnah. Thus, when they enter a masjid, none of them give Salaam to those present in it, due to their perceiving it as being makrooh. Perhaps, what we have written here may serve as a reminder for them as well as for others. And the reminder benefits the believer." 
In conclusion, the Salaam and the shaking of hands is to be done upon encountering for the first time and while parting, even if it is for a short while, and regardless if one is in the masjid or outside of it.
However, what is unfortunate, is that nowadays whenever you greet someone with "As-Salaam 'Alaikum wa Rahmatullaah" after prayer, upon your first encounter of him, he responds to you by saying: "Taqqabal Allaah". And he deems to have fulfilled what Allaah has obligated upon him from returning the greeting of Salaam. Thus, it is as if he has not heard Allaah's saying: "And when you are greeted with a greeting, then respond with something better than it (the initial greeting) or simply return its equal. Verily, Allaah takes account over all things." 
Some of these people even initiate the greeting by replacing the Salaam with the words "Taqabbal Allaah", and Allaah says: "Their greeting on the day they will meet Him will be Salaam." 
Also the Prophet (sallAllaahu 'alayhi wa sallam) said: "Spread the greeting of Salaam amongst yourselves."  And he did not say: "Say:Taqqabal Allaah"!!
And we do not know of anyone among the Sahaabah or the pious predecessors that, upon completion of the prayer, would turn to their right and left, shaking hands with those who were around him, congratulating them with the acceptance of the prayer. And if anyone of them had done that, then indeed it would have been conveyed to us in some report, even with a weak chain of narration. And our scholars would have quoted them for us, for they are the ones that are deeply engrossed with knowledge on every subject and submerged with its profundity. And they are the ones who derive the rulings from the ahaadeeth. Furthermore, they (all the scholars together) are not neglectful of any aspect of the Sunnah, whether it occurred in speech, action, silent approval or in a description. 
How could this be so, when our scholars of today have reported that this form of shaking hands, under those circumstances, is an innovation?!
Al-'Izz Ibn'Abd-is-Salaam said: "Shaking hands after the Fajr and 'Asr prayers is an innovation, unless it is done by one who arrives before the prayer and shakes hands with those that are gathered there, for indeed shaking hands is prescribed during the state of encountering. The Prophet (sallAllaahu 'alayhi wa sallam) would instead (of shaking hands), observe the remembrances that have been reported in the ahaadeeth, after completing his prayer. He would seek Allaah's forgiveness three times and then stop!! And it has been reported that he used to say: 'My Lord, protect me from Your punishment on the Day when Your servants will beresurrected." 
And all goodness lies in following the way of the Messenger of Allaah (sallAllaahu 'alayhi wa sallam)." 
If this innovation was confined to only the Fajr and 'Asr prayers during the lifetime of this author, then indeed it has grown, in our time, to the extent that it o ccurs after every one of the prayers! And there is no might or power except by the Will of Allaah!
Al-Lucknowee said: "In our time, this has spread throughout many lands, especially the land of Dakkan, which adheres to much innovations and afflictions. There are two matters that everyone must refrain from doing:
The First: When the people enter the masjids at the time of Fajr prayer, they do not greet one another with Salaam. Rather, they enter, pray the Sunnah prayer, then the obligatory prayer and then they greet one another with Salaam, after having completed the prayer and fulfilled its duties. This is reprehensible, for indeed the Salaamis only that which is given upon encountering and meeting one another, as has been confirmed in the reports, not while the people have already been sitting in a gathering, prior to that.
The Second: They shake hands with one another after completing the Fajr and 'Asr prayers and the two 'Eed and Jumu'ah prayers. This is in spite of the prescription that shaking hands is to be done upon first meeting one another." 
Then he stated after mentioning the difference of opinion concerning the shaking of hands after the prayer: "From those that forbade it was Ibn Hajr Al-Haithamee Ash-Shaafi'ee (rahimahullaah and Qutb-ud-Deen Ibn 'Alaa-ud-Deen Al-Makkee al-Hanafee (rahimahullaah). Also Al-Faadil Ar-Roomee in his book Majaalis-ul-Abraarhas placed it among the reprehensible acts of innovation, such that he said: 'Shaking hands is good when done in the state of encountering. As for when it is done in a state other than that of encountering, such as its being done (specifically) after the Jumu'ah and 'Eed prayers, as is the custom in our time, then there is no mention of it in the hadeeth. Thus it rem ains without any evidence. And we have already expounded before, that anything that does no have evidence to support it, is to be rejected. And blindly following in the practice of it is not permissible.'" 
He (rahimahullaah) also said: "The scholars of Fiqh from the Hanafee, Shaafi'ee and Maalikee madhhabs have all indicated that it is abhorred and that it is an innovation. It is stated in Al-Multaqit: 'Shaking hands (specifically) after prayer is detestable under every condition, because the Sahaabah never shook hands after the prayer and because it is from the practices of the Raafidah. Ibn Hajr (Al-Haithamee), from the Shaafi'ee scholars, said: 'What the people do after completing the five daily prayers, such as shaking hands, is detested. There is no source for it in the Religion.' " 
Then he (rahimahullaah) declared his conclusion and his preference in this matter, bysaying: "As for what I hold, then: Indeed they (the scholars) all agreed that this form of shaking hands has no basis in the Religion. Then they differed as to whether it was detested or allowed. So when a matter lingers between the ruling of makrooh (detested) and mubaah(allowed), it necessitates that it's ruling should be that it must be restricted. This is because stopping a harm takes precedence over bringing about good.  So how can it not take precedence over doing an allowed (mubaah) matter, when those that call to the shaking of hands, in our time, think that it is a good thing and repudiate severely anyone that forbids it and persist intensely on practicing it? And it is well known that persisting and going to excessiveness in matters that are recommended, takes those matters to the level of detestability. So how is it inregards to this matter, when there is a persisting being done upon an innovation that has no basis to it in the Religion?! Thus, according to this, there is no doubt that it is detested. And this was the aim behind those that ruled in favor of its detestability along with its ruling being reported from the explanations of the scholars of the past. Thus, it can not be opposed by the reports found in the books such as Jam'-ul-Barakaat, As-Siraaj-ul-Muneer and Mataalib-ul-Mu'mineen, since their authors' little concern for the authenticity of the narrations held within them, is a well known matter amongst the majority of the scholars. And it is amazing to see how the author of the book Khazaanat-ur-Riwaayahhas included in his work that the Prophet (sallAllaahu 'alayhi wa sallam) said: 'Shake hands after the Fajr prayer, Allaah will record ten good deeds for you due to it.' And that he (sallAllaahu 'alayhi wa sallam) said: 'Shake hands after the 'Asr prayer, you will be rewarded with mercy and forgiveness.' And it is not hard to understand that these two hadeeths and their likes are fabricated lies, which those that aspire to institute shaking hands (specifically after the prayer) have forged. Indeed to Allaah do we belong and indeed to Him We will return." 
Lastly, it must be noted that it is not permissible for a Muslim to interrupt the tasbeeh of his Muslim brother, unless there is a legitimate reason for it. Nor (is it permissible) what we have witnessed from the pestering of many Muslims, when they are observing the legislated remembrances after the congregational prayers, that they are confronted suddenly by hands that are extended in their direction from the left and the right for shaking, due to what it brings about from vexation and annoyance. This act is not permissible, not due to the aspect of shaking hands, but rather because it disrupts the people from making tasbeeh and preoccupying themselves with the remembrance of Allaah (dhikr).
So even if it is this way, then it is not wise for you to withdraw your hand from the hand of your neighbor and to reject accepting the hand that is extended out to you, for indeed this is from rudeness. And Islam does not tolerate such behavior. Rather, you should take hold of his hand with gentleness and kindness, and inform him of the innovative status of this type of handshake, which the people have introduced. And how many people have taken heed from a lesson and were receptive of the advice? Surely, it is only ignorance that has caused him to fall into the opposition of the Sunnah. Thus, it is for the scholars and their students to explain this to the people in a good manner. And perhaps it is the case that a man or a student of knowledge may desire to forbid an evil, but he is not able to conduct the correct and proper manner of doing it. And so he falls into an evil, which is greater than the evil he desired to forsake in the first place. So observe gentleness O Callers of Islam!! And cause the people to love you, by showing them that you possess good manners. In this way, you will be able to reach into their hearts. And you will find that they will lend you attentive ears and that they will have conscious hearts. For indeed, it is from the inner nature of mankind to run away from harshness and severity. 
2. Prostrating to Supplicate after the Prayer is Over:
There has come to pass a custom of the people, in which they prostrate themselves in supplication, after the prayer is over. And it is not known if this extra prostration has any basis to it, for it has not been reported on the Prophet (sallAllaahu 'alayhi wa sallam) or any of his companions. Rather, it is best to supplicate during one's prayer, based on the established reports regarding that. This is what the author of the book An-Nutmahhas stated. And Abu Shaamah quotes him, saying: "The act of the prostrating -which serves as a way of getting close to Allaah -duringprayer, does not necessitate that it serves the same purpose outside of prayer, as is the case with bowing."
Al-'Izz Ibn'Abd-is-Salaam (rahimahullaah) said: "The Divine Law does not mention that one can attain nearness to Allah with a particularized prostration. And there is no purpose in doing that, for indeed, the nearness that is gained from it, has causes, conditions, time constraints and foundations, of which it would notbe valid in their absence. So just as one does not draw close to Allaah by stopping at 'Arafah and Muzdalifah, casting the pebbles and walking around As-Safaa and Al-Marwa, without having slaughtered a sacrifice in its due time along with its causes and conditions, then similarly, one does not draw close to Allaah with a particularized prostration. And this is in spite of it's being an act of worship, unless there is a valid reason to do it. Likewise, one can not draw close to Allaah by observing fast and prayer at every place and during every time." 
In conclusion: This Religion does not mention anything about drawing closer to Allaah with the prostration, except in the prayer or due to a specific valid reason, such as forgetfulness (sujood-us-sahw), thankfulness or when reciting an ayah of prostration.
Abul-Mu'aalee, the Imaam of the Haramain, Al-Ghazaalee and others have forsaken these two prostrations. In fact, Al-Ghazaalee said: "No one held the opinion that the prostration, by itself, required a separate vow." 
The origin of this innovation comes from what some of the Sufis held in that it is preferable for every one that is praying to do that after the prayer is over, in order to coerce oneself, due to the heart's forgetfulness while in prayer. This is since no heart is void of forgetfulness during prayer, even if it is just for a moment. And they held that forgetfulness for the most part came from the Devil. Thus one should not reprimand himself, except with an attribute that the Devil cannot approach the servant in.
And there is no doubt that the Devil is the one that has indeed whispered this innovation to them, by causing them to have love for innovating into the Religion. And since the essence of the prayer is to strictly follow (what is stated in the texts), the scholars have ruled that this is an innovation.
3. Making Tasbeeh and Du'aa in Congregation while Disturbing those Praying:
It is not from the Sunnah for people to sit together after the prayer in order to recite some remembrances (dhikr) or supplications (du'aa) - those that have been reported or those that have not been reported - by raising the voice and joining in unison. This act has become the normal custom in some lands, even to the point that this tradition has become accepted, by the people, as being one of the trademarks of the Religion. It is such that anyone who abandons it or forbids others from doing it is renounced, whereas renouncing that it be abandoned is what is truly evil.
The author of As-Sunan wal-Mubtadi'aat said: "Supplicating to Allaah for forgiveness in congregation, in one unified voice, after finishing the prayer is an innovation. Likewise, their saying after the supplication: "O Most Merciful of those who show mercy, have mercy on us" (Yaa Arhamar-Raahimeen Arhamnaa) in congregation is an innovation. Connecting the optional prayer with the obligatory prayer without there being a period of division between the two is forbidden, as occurs in the hadeeth of Muslim. Reciting Al-Faatihahan extra time in honor of the Prophet is an innovation. The people's gathering together, after the morning prayer is finished, in order to say: "O Allaah, save me from the Hellfire" seven times is an innovation. Similarly, their adding after this: "… and from the torment of the Hellfire, by Your grace, O All-Mighty, O All-Forgiving" is an innovation." 
Ash-Shaatibee (rahimahullaah) said: "The Prophet (sallAllaahu 'alayhi wa sallam) did not used to say the supplication and the remembrances out loud immediately after prayer, all the time. Nor did he openly demonstrate doing it to the people, other than a time when it was done for the purpose of educating. This is because if he ha d done it and openly displayed it all of the time, it would have been from the Sunnahand the scholars would not have said that it is not from the Sunnah to do it, since its particularities, according to what they have stated, consist of its continuity an dits being done out loud amidst the congregation of people. And it should not be said: 'Had his supplication been done silently, it would nothave been memorized from him', for we will respond: 'Whoever is accustomed to doing it silently, then it will definitely be exhibited from him, either because of its habit or because of his efforts to warn about its becoming obligatory." 
This is as has been reported in Saheeh Al-Bukhaaree on Ibn 'Abbaas that he (radyAllaahu ‘anhu)) said: "Indeed, raising the voice whilst reciting dhikr, when the people would finish performing the congregational prayers, was done at the time of the Prophet (sallAllaahu 'alayhi wa sallam)." 
Imaam An-Nawawee (rahimahullaah) said: "Ash-Shaafi'ee understood this hadeeth to mean that they would raise their voices with it for a short period of time, for the purpose of honoring the attribute of dhikr. It did not mean that they would say it out loud all the time. The best opinion is that the Imaam and the ones following him in prayer should recite the dhikrin low voices (after prayer), unless there is aneed to educate others (by saying it loudly)." 
Ibn Bataal said: "It is stated in Al-'Utbiyyahthat Maalik (rahimahullaah) considered that to be an innovation."  Ash-Shaatibee (rahimahullaah) said: "It is concluded that making du'aain the form of congregation, all the time, was not from the actions of the Messenger of Allaah rnor from his statements or silent approvals." 
Ibn Al-Qayyim (rahimahullaah) said: "As for making du'aa (supplication) after finishing the prayer, whether facing the Qiblah or facing the followers in prayer, then this was not from his (sallAllaahu 'alayhi wa sallam) guidance at all. Nor has this action been reported on him whether through a saheeh or hasan chain of narration. And as for specifying that to be done in the Fajr and 'Asr prayers, then he never did this nor did any of his Khaleefahs. Nor did his ummah direct towards its implementation. Rather, it was s imply an opinion that was put into practice, by those that held that it was to take the place after the two prayers. And as for the supplications that are connected with the prayer, then indeed he (sallAllaahu 'alayhi wa sallam) used to say them while in his prayer and he commanded others to do that also. This is what is most befitting based on the condition of the one praying, for indeed when one prays, he is facing his Lord and conversing privately with Him, so long as he remains in prayer. But when he recites the tasleem, that private counsel comes to an end and his position of being in the presence of his Lord and his nearness to Him ceases. So how is it that he can abandon asking Him while he is in the state of private counsel with Him, nearness to Him and responsiveness, and then ask Him when he finishes from performing his prayer?! There is no doubt that the opposite of this condition is better for the one praying." 
One should make Istighfaar (i.e. say Astaghfirullaah) three times and then say the tasbeeh, the tahmeed, and the takbeer thirty three times each and then finish that with the tahleel, after the prayer. One should observe quietness in any state he is in after the prayer, such as standing, sitting or walking. And indeed, gathering to perform that (dhikr after prayer), participating in it and raising the voice out loud whilst doing it, are all innovations. The fact that it has become a habit seems insignificant to the people. And if anyone should call them to apply these similar characteristics to any other of the acts of worship, such as the prayer for entering the masjid (two raka'aat) for example, then they would renounce him, in the utmost manner. 
Also from its types is what has been innovated from making dhikr after each of the two Salaams at the end of the night prayer in Ramadaan, as well as doing it in a loud voice and in one unified rhythm. For indeed that is from the innovations.
1. A section of his book Al-Fasl-ul-Mubeen fee Akhtaa il-Musalleen
2. Reported by Abu Dawood in his Sunan (no. 5200) with anauthentic chain. All of its narrators are reliable. See Silsilat-ul-Ahaadeeth As-Saheehah no. 186Reported by Abu Dawood in his Sunan (no. 5200) with anauthentic chain. All of its narrators are reliable. See Silsilat-ul-Ahaadeeth As-Saheehah (no. 186)
3. Reported by Abu Dawood in his Sunan (no.927) and Ahmad in his Musnad (no. 2/30) with an authentic chain of narration according to the conditions of Al-Bukhaaree and Muslim. See As-Saheehah (no. 185)
4. Masaa'il Al-Marwazee (pg. 22)
5. 'Aaridat-ul-Ahwadhee Sharh Sunan At-Tirmidhee (2/162)
6. Reported by Al-Bukhaaree and Muslim
7. Nazl-ul-Abraar (pg. 350-351)
8. Silsilat-ul-Ahaadeeth As-Saheehah (1/314)
9. Surat-un-Nisaa: 86
10. Surat-ul-Ahzaab: 44
11. Reported by Muslim in his Saheeh (no. 54) and Ahmad in Al-Musnad (2/391, 442, 447) and others
12. See Tamaam-ul-Kalaam FeeBida'iyyat-il-Musaafaha Ba'ada As-Salaam (pg. 24-25)
13. The use of the author's (Al-'Izz) words: "It has been reported" indicates that the hadeeth is weak. However, it is not so, thus it would have been better for him to have said: "It is established". The hadeeth is found in Saheeh Muslim (no. 62), Sunan At-Tirmidhee and Musnad Ahmad.
14. Fataawaa Al-'Izz Ibn'Abd-is-Salaam (pg. 46-47); Also see Al-Majmoo' (3/488)
15. As-Si'aayah fee Al-Kashfi 'amaa fee Sharh Al-Wiqaayah (pg. 264). From his words it is understood that there is no harm in the shaking of hands between two or more people that did not meet each other, prior to that. Al-Albaanee said in As-Saheehah (1/23): "As for shaking the hands specifically after the congregational prayers, then this is an innovation of which there is no doubt, unless it occurs between two people that did not encounter each other, prior to that. In this case, it would be recommended for them to shake hands."
16. As-Si'aayah fee Al-Kashfi 'amaa fee Sharh Al-Wiqaayah (pg.264). Also see Ad-Deen-ul-Khaalis (4/314) and Al-Madkhal (2/84)
17. As-Si'aayah fee Al-Kashfi 'amaa fee Sharh Al-Wiqaayah (pg. 264)
18. Editor's Note: This is since stopping a harm will bring about good, whereas bringing about good does not necessarily stop a harm.
19. As-Si'aayah fee Al-Kashfi 'amaa fee Sharh Al-Wiqaayah (pg.265)
20. See Tamaam-ul-Kalaam Fee Bida'iyyat-il-Musaafaha Ba'ada As-Salaam (pg. 23)
21. Al-Baa'ith 'Alaa Inkaar-il-Bid'i wal-Hawaadith (pg. 58)
22. Musaajala 'Ilmiyyah (pg. 7-8)
23. See Al-Baa'ith 'Alaa Inkaar-il-Bid'i wal-Hawaadith (pg. 57-58), Al-Wajeez (2/234) and Al-Masjid feel-Islaam (pg. 227-228)
24. As-Sunan wal-Mubtadi'aat (pg. 70)
25. Al-'Itisaam (1/351)
26. Saheeh Al-Bukhaaree (no. 841) and Sunan Abu Dawood (no. 1002)
27. As is reported fromhim in Fath-ul-Baaree (2/326)
31. See Fataawaa Muhammad Rasheed Ridaa (4/1358-1359)
Source: Al-Fasl-ul-Mubeen Fee Akhtaa-il-Musalleen