The taslim

It is narrated from Sayyiduna Abdullah bin Zayd (may Allah be pleased with him) that the Prophet (peace and blessings be upon him) said: "The opening of the prayer is purification and ablution; its tahrim (that is, the act by which one enters the prayer and makes worldly matters forbidden upon oneself) is declaring the greatness of Allah (saying the takbir); and its tahlil (that is, the act by which the restriction of the prayer ends and other things become permissible) is saying the salam."
Al-Harith recorded this hadith in his Musnad (Bughyat al-Bahith). Its chain contains Muhammad bin Umar al-Waqidi, who is a da'if (weak) narrator; however, the text of this hadith is strengthened by other authentic reports and is reliable from a scholarly standpoint.

It is narrated from Ata bin Yasar (may Allah have mercy on him) that he said: "Indeed, the Prophet (peace and blessings be upon him) used to say the salam only once, to his right side (on finishing the prayer)."
Ibn Abi Shaybah transmitted this report in his al-Musannaf, and this hadith is the evidence of those jurists who consider a single salam at the end of the prayer sufficient, although the hadith scholars have criticised it because the narrator Muhammad bin Umar al-Waqidi is alone in its chain.

It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both): "He (on completing the prayer) said the salam only once, while still facing towards the qiblah."
Ibn Sa'd transmitted this report in his book al-Tabaqat al-Kubra. The narrator in its chain, Muhammad bin Umar al-Waqidi, is da'if (weak) according to the majority of hadith scholars; however, the practice of saying the salam to one side in prayer is established by other authentic reports and supporting evidence.

It is narrated from Imam al-Zuhri (may Allah have mercy on him) that he saw Qabisah bin Dhu'ayb, when he said the salam of the prayer, saying the salam only once while still facing towards the qiblah. Imam al-Zuhri (may Allah have mercy on him) said: I mentioned this to Abdullah bin Mawhab, and he told me: I asked Qabisah about this, and he replied: "I saw Sayyiduna Zayd bin Thabit (may Allah be pleased with him) saying the salam only once (at the end of the prayer) while still facing towards the qiblah."
Ibn Abi Shaybah recorded it in al-Musannaf (Hadith: 3108), and its chain contains the narrator al-Waqidi, who is da'if (weak).

Sa'id bin Ata bin Abi Marwan al-Aslami narrates from his father, from his grandfather, that he said: "I prayed behind Sayyiduna Umar, Sayyiduna Ali and Sayyiduna Abu Dharr (may Allah be pleased with them), and I saw all of them saying the salam (at the end of the prayer) to their right side and to their left side."
[Takhrij: Al-Bayhaqi recorded this athar in al-Sunan al-Kubra (2921). Its chain contains Muhammad bin Umar al-Waqidi, a da'if (weak) narrator; however, the basis of saying the salam to both sides in prayer is established by other authentic hadith and reports.]

It is narrated from Sayyiduna Uqbah bin Amir (may Allah be pleased with him) that he said: "I saw the Messenger of Allah (peace and blessings be upon him) at the end of the prayer saying the salam to his right side and to his left side, uttering these words: 'As-salamu 'alaykum wa rahmatullah, as-salamu 'alaykum wa rahmatullah' (Peace be upon you and the mercy of Allah; peace be upon you and the mercy of Allah)."
The chain of this report contains Muhammad bin Umar al-Waqidi, who is da'if (weak) according to the hadith scholars; however, saying the salam to both the right and the left in prayer is a pure Sunnah established by other authentic and reliable hadith.

It is narrated from Sayyiduna Ali (may Allah be pleased with him): "He said the salam (at the end of the prayer) to his right and to his left, and then he stood up (from his place of prayer)."
Abu Bakr bin Abi Shaybah narrated this athar with a reliable chain.

It is narrated from Sayyiduna al-Bara' bin Azib (may Allah be pleased with him) that the Prophet (peace and blessings be upon him) used to say the salam to his right and to his left, uttering the words 'As-salamu 'alaykum wa rahmatullah' (Peace and the mercy of Allah be upon you), until the whiteness of his (peace and blessings be upon him) radiant cheek could be seen (by the person sitting behind, because he turned his blessed neck).
Ibn Abi Shaybah narrated this hadith (Hadith: 532). The narrator in its chain, Hurayth bin Abi Matar, is da'if (weak); however, the basic Sunnah of saying the salam to both sides in prayer is established by other reliable hadith, including the narration of Sahih Muslim (582).

Harithah bin Mudarrib (may Allah have mercy on him) said: "Sayyiduna Ammar bin Yasir (may Allah be pleased with him) was our governor for one year, and in that period, whatever prayer he led us in, at its end he would always say the salam to his right and to his left, uttering these words: 'As-salamu 'alaykum wa rahmatullah, as-salamu 'alaykum wa rahmatullah' (The peace of Allah and His mercy be upon you; the peace of Allah and His mercy be upon you)."
The chain of this hadith is sahih (authentic), and Musaddad bin Musarhad recorded it in his Musnad.