The tashahhud in prayer

Sayyiduna Umar bin al-Khattab (may Allah be pleased with him), while delivering the sermon from the pulpit of the Messenger of Allah (peace and blessings be upon him), used to teach the people the tashahhud of the prayer, saying: When one of you recites the tashahhud, let him say: 'Bismillahi khayril-asma', at-tahiyyatuz-zakiyatus-salawatut-tayyibatu lillah, as-salamu 'alayka ayyuhan-Nabiyyu wa rahmatullahi wa barakatuh, as-salamu 'alayna wa 'ala 'ibadillahis-salihin, ashhadu an la ilaha illallahu wahdahu la sharika lahu wa anna Muhammadan 'abduhu wa rasuluh' (In the name of Allah, the best of names. All verbal, pure, bodily and financial worship is for Allah alone. Peace be upon you, O Prophet, and the mercy of Allah and His blessings. Peace be upon us and upon the righteous servants of Allah. I bear witness that there is no god but Allah, alone, with no partner, and that Muhammad (peace and blessings be upon him) is His servant and Messenger). Sayyiduna Umar also said: After sending salam upon the Messenger of Allah (peace and blessings be upon him), begin with yourselves, and then send peace upon the righteous servants of Allah.
This hadith is sahih on the condition of Muslim, and I have mentioned it because it has supporting narrations (shawahid).

Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) took the hand of Awn bin Abdullah and taught him the words of the tashahhud, and said: Sayyiduna Umar bin al-Khattab (may Allah be pleased with him) took my hand and said that the Messenger of Allah (peace and blessings be upon him) had taken his hand and taught him this tashahhud: 'At-tahiyyatus-salawatut-tayyibatuz-zakiyatu lillah' (All verbal, bodily, financial and pure worship is for Allah).
The mention of Allah's name at the beginning of the tashahhud is sahih and in accordance with the condition of Imam Bukhari.

Sayyiduna Jabir (may Allah be pleased with him) relates that the Messenger of Allah (peace and blessings be upon him) used to teach us the tashahhud just as he taught us a surah of the Qur'an: 'Bismillahi wa billah, at-tahiyyatu lillah' (In the name of Allah and by the help of Allah; all verbal worship is for Allah). And at its end was this supplication: 'Allahumma inni as'alukal-jannata wa a'udhu bika minan-nar' (O Allah, I ask You for Paradise and I seek refuge in You from the Fire).

Sayyiduna Jabir bin Abdullah (may Allah be pleased with them both) relates that the Messenger of Allah (peace and blessings be upon him) used to teach us the tashahhud just as he taught us a surah of the Qur'an: 'Bismillahi wa billah, at-tahiyyatu lillah, as-salawatut-tayyibatu lillah, as-salamu 'alayka ayyuhan-Nabiyyu wa rahmatullahi wa barakatuh, as-salamu 'alayna wa 'ala 'ibadillahis-salihin, ashhadu an la ilaha illallah wa ashhadu anna Muhammadan 'abduhu wa rasuluh, nas'alullahal-jannata wa na'udhu bihi minan-nar' (In the name of Allah and by the grace of Allah. All verbal, bodily and financial worship is for Allah. Peace be upon you, O Prophet, and the mercy of Allah and His blessings. Peace be upon us and upon the righteous servants of Allah. I bear witness that there is no god but Allah, and that Muhammad (peace and blessings be upon him) is the servant and Messenger of Allah. We ask Allah for Paradise and seek refuge in Him from the Fire).
Imam al-Hakim says: Ayman bin Nabil is trustworthy (thiqah), and Imam Bukhari used him as evidence; Yahya bin Ma'in also declared him trustworthy.

A similar narration is reported from Sayyiduna Jabir (may Allah be pleased with him), which is sahih on the condition of Imam Muslim.
Imam al-Hakim says: A narrator in its chain made an error, but the original text is the one mentioned earlier.

Sayyiduna Mihjan bin al-Adra' (may Allah be pleased with him) relates that the Messenger of Allah (peace and blessings be upon him) entered the mosque, and there was a man who, having prayed, was making this supplication in the tashahhud: 'Allahumma inni as'aluka billahil-Ahadis-Samad, alladhi lam yalid wa lam yulad, wa lam yakun lahu kufuwan ahad, an taghfira li dhunubi, innaka antal-Ghafurur-Rahim' (O Allah, I ask You by virtue of Your being One and Self-Sufficient, the One who neither begot nor was begotten, and to whom no one is equal, that You forgive my sins; indeed You alone are the Most Forgiving, the Most Merciful). On hearing this, he (peace and blessings be upon him) said three times: "He has been forgiven, he has been forgiven, he has been forgiven."
This hadith is sahih on the condition of the two Shaykhs, but they did not record it.

Sayyiduna Abdullah bin Mas'ud (may Allah be pleased with him) relates: It is the Sunnah that the tashahhud be recited quietly (silently).
This hadith is sahih on the condition of Muslim, but the two Shaykhs did not record it.

It is narrated from Sayyiduna Abu Mas'ud Uqbah bin Amr (may Allah be pleased with him) that a man came and sat in front of the Messenger of Allah (peace and blessings be upon him) while we were with him, and said: O Messenger of Allah, as for sending salam upon you, we have learned (how to do) it, but when we send blessings upon you in our prayers, how should we send blessings upon you, may Allah send His blessings upon you? The narrator says: He (peace and blessings be upon him) remained silent until we wished that the man had not asked him this question. Then he (peace and blessings be upon him) said: "When you send blessings upon me, say: 'Allahumma salli 'ala Muhammadin an-Nabiyyil-ummiyyi wa 'ala ali Muhammad, kama sallayta 'ala Ibrahima wa 'ala ali Ibrahim, wa barik 'ala Muhammadin an-Nabiyyil-ummiyyi wa 'ala ali Muhammad, kama barakta 'ala Ibrahima wa 'ala ali Ibrahim, innaka Hamidun Majid' (O Allah, send Your mercy upon Muhammad (peace and blessings be upon him), the unlettered Prophet, and upon his family, as You sent Your mercy upon Ibrahim (peace be upon him) and upon his family; and send Your blessings upon Muhammad (peace and blessings be upon him), the unlettered Prophet, and upon his family, as You sent Your blessings upon Ibrahim (peace be upon him) and upon his family; indeed You alone are Praiseworthy and Most Glorious)."
This hadith is sahih on the condition of Muslim, but the two Shaykhs did not record it with the mention of sending blessings upon the Prophet (peace and blessings be upon him) in the prayers.