The supplications of the funeral prayer

It is narrated from Sayyiduna Abu Hurayrah (may Allah be pleased with him) that when the Messenger of Allah (peace and blessings be upon him) led a funeral prayer, he would make this supplication: 'Allahumma-ghfir li-hayyina wa mayyitina, wa shahidina wa gha'ibina, wa saghirina wa kabirina, wa dhakarina wa unthana. Allahumma man ahyaytahu minna fa-ahyihi 'ala al-Islam, wa man tawaffaytahu minna fa-tawaffahu 'ala al-iman' (O Allah, forgive our living and our dead, those of us present and those absent, our young and our old, and our males and our females. O Allah, whomever of us You keep alive, keep him alive upon Islam, and whomever You cause to die, cause him to die upon faith).
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it! And it has an authentic supporting narration (shahid) on the condition of Muslim, as follows:

It is reported from the Mother of the Believers, Sayyidah Aishah (may Allah be pleased with her), that she said: The (supplication in the) prayer of the Messenger of Allah (peace and blessings be upon him) over the deceased was that he would say: 'Allahumma-ghfir li-hayyina wa mayyitina, wa dhakarina wa unthana, wa shahidina wa gha'ibina wa saghirina wa kabirina. Allahumma man ahyaytahu minna fa-ahyihi 'ala al-Islam, wa man tawaffaytahu fa-tawaffahu 'ala al-iman' (O Allah, forgive our living and our dead, our males and our females, those of us present and those absent, and our young and our old. O Allah, whomever of us You keep alive, keep him alive upon Islam, and whomever You cause to die, cause him to die upon faith).

It is reported from Yazid bin Abdullah bin Rukanah that when the Messenger of Allah (peace and blessings be upon him) stood before a funeral to pray over it, he would make this supplication: 'Allahumma 'abduka wa-bnu amatika, ihtaja ila rahmatika, wa anta ghaniyyun 'an 'adhabihi, in kana muhsinan fa-zid fi ihsanihi, wa in kana musi'an fa-tajawaz 'anhu' (O Allah, this is Your servant and the son of Your maidservant; he is in need of Your mercy, and You have no need to punish him. If he was a doer of good, then increase his good deeds, and if he was a wrongdoer, then pardon him).
This chain is sahih (authentic), and Yazid bin Rukanah and his father Rukanah bin Abd Yazid were both Companions from Banu al-Muttalib bin Abd Manaf, but the two Shaykhs did not record it.

Shurahbil bin Sa'd narrates: I saw Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) lead us in a funeral prayer at al-Abwa'. He said the takbir, then recited Surah al-Fatihah aloud, then sent blessings upon the Prophet (peace and blessings be upon him), then made this supplication: 'Allahumma 'abduka wa-bnu 'abdika wa-bnu amatika, yashhadu an la ilaha illa anta, wahdaka la sharika laka, wa yashhadu anna Muhammadan 'abduka wa rasuluka, asbaha faqiran ila rahmatika, wa asbahta ghaniyyan 'an 'adhabihi, takhalla min al-dunya wa ahliha, in kana zakiyan fa-zakkihi, wa in kana mukhti'an fa-ghfir lahu, Allahumma la tahrimna ajrahu wa la tudillana ba'dahu' (O Allah, this is Your servant, the son of Your servant and the son of Your maidservant. He bears witness that there is no god but You, alone, without partner, and he bears witness that Muhammad (peace and blessings be upon him) is Your servant and Your Messenger. He has come to be in need of Your mercy, and You have no need to punish him. He has left the world and its people behind. If he was pure (righteous), then grant him further purity, and if he was a wrongdoer, then forgive him. O Allah, do not deprive us of his reward, and do not lead us astray after him). Then he said three more takbirs, then finished and said: "O people, I recited (Surah al-Fatihah) aloud only so that you would know that it is the Sunnah."
The two Shaykhs did not use Shurahbil bin Sa'd as evidence, and he is one of the Tabi'un of the people of Madinah. I have presented this hadith only as a supporting narration (shahid) for the hadiths we mentioned earlier, because they were brief and summarised, whereas this hadith is detailed and clear.

It is narrated from Sayyiduna Abdullah bin Abi Awfa (may Allah be pleased with him) that a daughter of his died, and he followed the funeral riding a mule, while women were wailing (in lamentation) over her. He said: "Whether you wail or do not wail, the Messenger of Allah (peace and blessings be upon him) has certainly forbidden wailing; however, whoever of you wishes may shed her tears." Then he led the funeral prayer and said four takbirs. After the fourth takbir he remained standing for as long as the interval between two takbirs, supplicating and seeking forgiveness for her (his daughter), and he said that the Messenger of Allah (peace and blessings be upon him) used to do likewise.
This hadith is sahih (authentic), but they did not record it, and Ibrahim bin Muslim al-Hajari has not been criticised with any sound proof.

Abu Umamah bin Sahl bin Hunayf, who was one of the elders and scholars of the Ansar and one of the sons of those who took part in Badr with the Messenger of Allah (peace and blessings be upon him), narrates that some men among the Companions of the Messenger of Allah (peace and blessings be upon him) informed him about the funeral prayer: "The imam says the takbir, then sends blessings upon the Prophet (peace and blessings be upon him), then in the next three takbirs he makes sincere supplication (for the deceased), then he says the salam quietly when finishing; and the Sunnah is that the one behind him does the same as his imam did." Al-Zuhri said: Abu Umamah told me this while Sa'id bin al-Musayyib was listening, and he did not object to it. Ibn Shihab said: I mentioned to Muhammad bin Suwayd this Sunnah of the funeral prayer reported by Abu Umamah, and he said: I too heard al-Dahhak bin Qays narrating from Habib bin Maslamah, about a funeral prayer he led, the same as what Abu Umamah told us.
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it, and there is no narration more authentic than this regarding a single salam in the funeral prayer. Its supporting narration (shahid) is the following hadith of Abu al-Anbas Sa'id bin Kathir:

It is narrated from Sayyiduna Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) led a funeral prayer, said four takbirs over it and said a single salam.
Saying a single salam in the funeral prayer is established through authentic narrations from Sayyiduna Ali bin Abi Talib, Sayyiduna Abdullah bin Umar, Sayyiduna Abdullah bin Abbas, Sayyiduna Jabir bin Abdullah, Sayyiduna Abdullah bin Abi Awfa and Sayyiduna Abu Hurayrah (may Allah be pleased with them), that they used to say only a single salam in the funeral prayer.