Which Ahadith and references from the scholars could be provided to answer this question?
Authentic Supplications (A'diyah) during the (final) Tashahhud:
Brief mention: There are various versions of reciting the Tashahhud, and here are three which are reported that the Prophet (ﷺ) uttered: Muslim (403a, 404) and Al-Bukhari (6265). Then, there are various versions of the Durud for the Prophet (ﷺ) after the Tashahhud, and here are four examples of which are reported: Muslim (405, 406a, 407) and Al-Bukhari (6358). Finally, you recite the Du'a before any other which is narrated in Muslim (588a), and underlined by the narration of An-Nasa'i (1310), that was classed Sahih:
The Messenger of Allah (ﷺ) said: "When one of you recites the Tashahhud, let him seek refuge with Allah (swt) from four things: From the torment of hell, from the torment of the grave, from the trials of life and death and from the evils of the Dajjal. Then let him pray for himself asking whatever he wants."
Subsequently, it is narrated:
[...] 'there was a man who had finished his prayer and he was reciting the Tashahhud. He said: Ya Allah, I ask of You! Ya Allah, as You are the One, the Only, the Self-Sufficient Master, Who begets not nor was He begotten, and there is None equal or comparable to Him, forgive me my sins, for You are the Oft-Forgiving, Most Merciful' (Allahumma inni as'aluka, Ya Allah - bi-annakal Wahid - Al-Ahadus-samad - Alladhi lam yalid wa lam yulad, wa lam yakul-lahu kufuwan Ahad, an-taghfirali dhunubi - Innaka antal Ghafurur-Rahim) - The Messenger of Allah (ﷺ) said (then): "He has been forgiven", 'three times'.
[...] a man [...] recited the Tashahhud, he supplicated, and in his supplication he said: 'Ya Allah, indeed I ask You since all praise is due to You, there is none worthy of worship but You, the Bestower, the Creator of the heavens and earth, O Possessor of majesty and honor, O Ever-living, O Eternal.' (Allahumma inni as'aluka bi-anna lakal-hamd, la ilaha illa ant, Al-Mannan(u) badi'us-samawati wal-ard - Ya dhal-Jalali wal Ikram - Ya Hayyu, Ya Qayyum) The Prophet (ﷺ) said: "Do you know what he has supplicated with?" 'They said': 'Allah (swt) and His Messenger know best.' 'He (ﷺ) said': "By the One in Whose Hand is my soul, he called upon Allah by His greatest Name, which, if He is called by it, He responds, and if He is asked by it, He gives."
Al-Mustadrak (1/583), which Al-Hakim graded Sahih and Imam Dhahabi agreed, and Shaykh Albani concurred in As-Sahihah (7/209-12 - #3411); and also graded the next version Hasan which is narrated in Ibn Majah (3858):
'Ya Allah, I ask You by virtue of the fact that all praise is due to You; none has the right to be worshiped but You alone, and You have no partner or associate, the Bestower, the Originator of the heavens and the earth, the Possessor of majesty and honor.' (Allahumma inni as'aluka bi-anna lakal hamd - La ilaha illa ant - wahdaka la sharika lak - Al-Mannan(u) badi'us-samawati wal-ard - Dhul-Jalali wal Ikram)
In At-Tirmidhi (3499), which Abu I'sa classed Hasan, it is narrated that the Prophet (ﷺ) said, after he was asked 'which supplication is most likely to be listened to':
"(During) the last part of the night, and at the end of the obligatory prayers."
After that, following supplications have been narrated:
The Messenger of Allah (ﷺ) said to a man: "What do you say during your Salah?" He said: 'The Tashahhud, then I ask Allah for Paradise, and I seek refuge with Him from Hell, but I do not understand what you and Mu'adh murmur (during Salah). He (ﷺ) said: "Our murmuring revolves around the same things."
"When you ask from Allah, ask for Al-Firdaws" (Allahumma inni asalukal Jannatul Firdaws wa a'udhu bika minan-nar)
'Ya Allah, I seek refuge with You from Kufr, poverty, and the torment of the grave. (Allahumma inni a'udhu bika min al-kufr, wal-faqr, wa a'dhabil qabr) Continue to recite them, O my son, for the Prophet of Allah (ﷺ) used to say this supplication at the end of the prayer.'
The Prophet (ﷺ) said in some of his prayers: "Ya Allah, give me an easy reckoning" (Allahumma hasibni hisaban yasira(n))
Al-Mustadrak (1/385) which Al-Hakim graded Sahih according to the conditions of Muslim, and Imam Dhahabi concurred
Allah's Messenger (ﷺ) used to invoke Allah in the prayer, saying: "Ya Allah, I seek refuge with You from the punishment of the grave, from the afflictions of the Dajjal, and from the afflictions of life and death. O Allah, I seek refuge with You from sins and from debt." (Allahumma inni a'udhu bika min a'dhabil qabr, wa a'udhu bika min fitnatil masihid-dajjal, wa a'udhu bika min fitnatil mahya wa fitnatil mamat - Allahumma inni a'udhu bika minal ma'thami wal maghram) Somebody said to him: 'Why do you so frequently seek refuge with Allah from being in debt?' The Prophet (ﷺ) replied: "A person in debt tells lies whenever he speaks, and breaks promises whenever he makes (them)."
Farwah bin Nawfal said: 'I said to A'ishah: Tell me of a supplication that the Messenger of Allah (ﷺ) used to say in his prayer. She said': 'Yes, the Messenger of Allah (ﷺ) used to say': "Ya Allah, I seek refuge with You from the evil of that which I have done, and the evil of that which I have not done." (Allahumma inni a'udhu bika min sharri ma a'milt, wa min sharri ma lam a'mal (b'ad))
"Ya Allah, I have done great injustice to myself and none except You forgives sins, so bestow on me a forgiveness from You, and Have Mercy on me, You are the Forgiver, the Merciful." (Allahumma inni dhalamtu nafsi dhulman kathiran, wa la yaghfirdh-dhunuba illa ant, faghfirli maghfiratan min i'ndak, war-hamni, Innaka antal Ghafurur-Rahim (wa Akramul Akramin))
The Messenger of Allah (ﷺ) said: "Never forget to say in every prayer: "My Rabb, help me to remember You, give thanks to You and worship You well" (Rabbi a'inni a'la dhikrika wa shukrika wa husni i'badatik)
Sunan an-Nasa'i (1303); and in Abu Dawud (1522) it is narrated starting with "Allahumma...", both Ahadith are graded Sahih | What is meant by "at the end of the prayer (Dibr as-Salah)" here is in the last part of the prayer before the Salam because 'dibr ash-shay' (lit. the end of a thing) is part of it. This is confirmed by the words in the report of an-Nasa'i: "in every prayer". Ibn al-Qayyim said in Zad al-Ma'ad (1/295): 'At the end of the prayer' may be understood as meaning before the Salam or after it. Our Shaykh [Ibn Taymiyyah] regarded it as more likely that it is before the Salam. I asked him about that and he said: 'Dibr kulli shay' (the end of everything) is part of it, like the dibr (rear end) of an animal.'
'The supplication that he (the Prophet ﷺ made very frequently is this)': "Ya Allah, grant us the good in this world and the good in the Hereafter and save us from the torment of Hell-Fire." (Allahumma rabbana atina fid-dunya hasanatan wa fil akhirati hasanatan wa qina a'dhaban-nar) [...] 'Whenever Anas had to supplicate, he made this very supplication, and whenever he (intended) to make another supplication, he (inserted) this very supplication in that.'
Abu Malik reported on the authority of his father that when a person embraced Islam, Allah's Messenger (ﷺ) used to teach him how to observe prayer and then commanded him to supplicate in these words:
"Ya Allah, grant me pardon, have mercy upon me, direct me to the path of righteousness, grant me protection and provide me sustenance" (Allahum-maghfirli, war-hamni, wahdini, wa a'fini, war-dhuqni)
Abu Musa, when he finished the prayer, would say: 'Ya Allah, forgive me my sins, and ease my affairs, and bless my sustenance' (Allahum-maghfirli dhanbi, wa yas-sir li a-mri, wa barik li fi rizqi)
Allah's Messenger (ﷺ) used to seek refuge with Allah from them (i.e. the evils) at the end of every prayer. The words are: "Ya Allah, I seek refuge with You from cowardice, and seek refuge with You from being brought back to a bad stage of old life and seek refuge with You from the afflictions of the world, and seek refuge with You from the punishments in the grave." (Allahumma inni a'udhu bika minal bukhl, wa a'udhu bika minal jubn, wa a'udhu bika min an uradda ila ar-dhalil u'mur, wa a'udhu bika min fitna-tid-dunya wa a'dhabil qabr)
'When I finished, I asked: Messenger of Allah, what is the comprehensive supplication? He (ﷺ) said': Say: "Ya Allah, I ask You for all good, both sooner and later, what I know of it and what I do not know. I seek refuge with You from all evil, both sooner and later, what I know of it and what I do not know. I ask You for the Garden and whatever words or actions bring one near to it. I seek refuge with You from the Fire and whatever words or actions bring one near to it. I ask You by what Muhammad asked You and I seek refuge from You by what Muhammad sought refuge from and whatever fate You have decreed for me, make its end right guidance."
Adab al-Mufrad (639), Ibn Majah (3846) and others, graded Sahih in As-Sahihah (4/56 - #1542), and it is mentioned that Ibn Mas'ud would frequently say this in his Tashhahud, and he said: 'No Prophet or righteous person ever made supplication, except that it is contained in this Du'a.' Quoted in Fath al-Bari (2/321)
"Ya Allah, by Your knowledge of the unseen and Your power over creation, keep me alive so long as You know that living is good for me and cause me to die when You know that death is better for me. Ya Allah, cause me to fear You in secret and in public. I ask You to make me true in speech in times of pleasure and of anger. I ask You to make me moderate in times of wealth and poverty. And I ask You for everlasting delight and joy that will never cease. I ask You to make me pleased with that which You have decreed and for an easy life after death. I ask You for the sweetness of looking upon Your face and a longing to meet You in a manner that does not entail a calamity that will bring about harm or a trial that will cause deviation. Ya Allah, beautify us with the adornment of faith and make us among those who guide and are rightly guided."
Right before saying the Taslim:
[...] and he (ﷺ) would then say between Tashahhud and the pronouncing of salutation: "Forgive me of the earlier and later, open and secret (sins), and that where I made transgression and that Thou knowest better than I. Thou art the First and the Last - There is no god, but Thee" (Allahum-maghfirli ma qad-damtu wa ma akh-khart, wa ma asrartu wa ma a'lant, wa ma asraft, wa ma anta a'lamu bihi minni - Antal Muqad-dimu wa antal Mu-akh-khiru - La ilaha illa ant)
And, in At-Tirmidhi (3423), which Abu I'sa classed Hasan Sahih, to say: (Allahum-maghfirli ma qad-damtu wa ma akh-khart, wa ma asrartu wa ma a'lant, anta i-lahi ilaha illa ant)
- Special mention:
It is narrated in Mu'jam al-Kabir (7982), to which Al-Haythami said in Majm'a az-Zawa'id (10/112): 'And his men are the men of Sahih, other than Zubayr ibn Khariq, and he is trustworthy'; rather, 'he is not strong' as per Ad-Daraqutni, who was quoted and as it was explained in Sahih Abu Dawud (2/159-160).
Furthermore, for the above, there is a witness: From the Hadith of Abu Ayyub al-Ansari, which is narrated in Mu'jam al-Kabir (3875), Mu'jam al-Awsat (4442), Mu'jam al-Saghir (610), Mujalasat wa Jawahirul I'lm (1447), and Al-Mustadrak (3/522) | Shaykh Al-Hilal commented in his Tahqiq of Ibn as-Sunni (1/167-9 - #117): 'The Hadith is Hasan in the second way, and the evidence is from the Hadith of Abu Ayyub'; for which Shaykh Albani was likely inclined (due to ample amount of narrations) to grade it Hasan in Sahih al-Jam'i (1266) and explained it in Ar-Rawd an-Nadir (910).
Additionally, Al-Haythami said in (10/173): 'At-Tabarani[*] narrated it, and his men are trustworthy', to which Ibn Hajar commented, while grading it with a Hasan Isnad in Mukhtasar Zawaid (2/433-4): 'And, he did not attribute it to Al-Bazzar, which is probably a misapprehension', who narrated it (with the same narrators sans Abu Ayyub) in his Kashf al-Astar (3192) and Musnad al-Bazzar (5997):
Ibn U'mar said: 'I did not pray behind the Prophet (ﷺ) except that I heard him saying, when he finished his prayer: "Ya Allah, forgive me my transgressions and what I did intentionally; Ya Allah, guide me to righteous deeds and good manners - because no one guides to good manners and no one averts evil, but You" (Allahum-maghfirli khatayaya wa a'mdi; Allahum-mahdini li-salihil a'mali wal akhlaq - Innahu la yahdi li-salihiha wa la yasrifu say-yi-a-ha illa ant)