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 which is underlined in the narration of An-Nasa'i (1310), which is graded 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'. Narrated in Sunan an-Nasa'i (1301) graded Sahih
[...] 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." Narrated in Al-Hakim (1/583) which was graded Sahih and Imam Dhahabi agreed, and Shaykh Albani in As-Sahihah (7/209-12 - #3411); and in another version narrated in At-Tirmidhi (3544): 'Ya Allah, none has the right to be worshipped but Allah, You are the One Who gives blessings, Originator of the heavens and the earth, Possessor of glory and generosity' (Allahumma la ilaha illa ant - Al-Mannan(u) badi'us-samawati wal ard - Dhal Jalali wal Ikram); and another one in Ibn Majah (3858), which Shaykh Albani graded Hasan, it was narrated: '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." Ibn Majah (910) graded Sahih as well as in Abu Dawud (792); in At-Tirmidhi (2530) and Musnad Ahmad (22695, 22738), which was graded Sahih, it is narrated that the Prophet (ﷺ) said: "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.' Narrated in Sunan an-Nasa'i (1347, 5465) graded with a Sahih Isnad by Shaykh Albani
The Prophet (ﷺ) said in some of his prayers: "Ya Allah, give me an easy reckoning." (Allahumma hasibni hisaban yasira(n)) Narrated in Al-Hakim (1/385) which was graded Sahih and Imam Dhahabi agreed
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)." Al-Bukhari (832-3)
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)) Narrated in Sunan an-Nasa'i (1307, 5523-8) graded Sahih; Muslim (2716a-b, d)
"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) Al-Bukhari (834)
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) Narrated in 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.' Muslim (2690a), Al-Bukhari (4522, 6389)) - 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) Muslim (2697b)
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) Narrated in Ibn Abi Shaybah (3033, 29255) Sahih Isnad according to Shaykh Albani in Tamam al-Manat (p. 96)
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) Al-Bukhari (2822, 6390)
'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 by Shaykh Albani in As-Sahihah (4/56 - #1542), and it is mentioned that Ibn Mas'ud would frequently say this in his Tashhahud, and he mentioned: 'No Prophet or righteous person ever made supplication, except that it is contained in this Du'a.' Quoted in Fath al-Bari (2/321)
Narrated Abdullah ibn Mas'ud: 'We did not know what we should say when we sat during prayer. The Messenger of Allah (ﷺ) was taught (by Allah). [...] He said: He used to teach us also some other words, but he did not teach them as he taught us the tashahhud': 'O Allah, join our hearts, mend our social relationship, guide us to the path of peace, bring us from darkness to light, save us from obscenities, outward or inward, and bless our ears, our eyes, our hearts, our wives, our children, and relent toward us; Thou art the Relenting, the Merciful. And make us grateful for Thy blessing and make us praise it while accepting it and give it to us in full.' Narrated in Abu Dawud (969), in Adab al-Mufrad (630), etc. and Shaykh Albani graded it Sahih
"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." Narrated in Sunan an-Nasa'i (1305-6) graded Sahih in Sahih al-Jam'i (1301); there are different, shorter versions like in the narrations related in At-Tirmidhi (3235), which Abu I'sa classed Hasan Sahih and he added: 'I asked Muhammad bin Isma'il (Al-Bukhari) and he said: 'This hadith is Hasan Sahih' - "(Allah) said: "Ask!" (Muhammed ﷺ) said: "Ya Allah, I ask of you the doing of the good deeds, avoiding the evil deeds, loving the poor, and that You forgive me, and have mercy upon me. And when You have willed Fitnah in the people, then take me without the Fitnah. And I ask You for Your love, the love of whomever You love, and the love of the deeds that bring one nearer to Your love." (Allahumma inni asaluka fi'lal khayrat, wa tarkal munkarat, wa hub-bal masakin, wa an taghfira li wa tar-hamni wa idha a-rad-ta fit-nata-qaumin fatawaf-fani ghayra maftun, asaluka hub-baka wa hub-ba man yuhib-buka wa hub-ba a'malin yuqar-ribu ila hub-bik), and the other two (3233-4) (Allahumma inni asaluka fi'lal khayrat, wa tarkal munkarat, wa hub-bal masakin, wa idha a-radta bi-i'badika fitnatan, faqbidni ilayka ghayra maftun) which Shaykh Albani classed Sahih li Ghayrihi in Sahih at-Targhib (194, 302, 408, 451, 3192)
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) Muslim (771a) and in (771b) it is stated that he (ﷺ) did not (just) finish with this Du'a.
- Special mention:
And it is narrated in Mu'jam al-Kabir (7811) and others, which Shaykh Albani classed Hasan in Sahih al-Jam'i (1266) without comment, and the scholars had different opinions about the narrator Zubayr ibn Khariq (an analysis has been made here):
Abu Umamah said: 'I never came near the prayer of the Messenger of Allah (ﷺ) except that I heard him say at the end of an obligatory and voluntary prayer': "Ya Allah, forgive me my sins as well as my transgressions - Ya Allah, revive, enrich and guide me to righteous deeds and morals because no one guides to morals and no one averts evil, but You" (Allahum-maghfirli dhunubi wa khatayaya - Allahumma an-i'shni waj-birni wah-dini li-salihil a'mali wal akhlaq, fa-innahu la yahdi li-salihiha wa la yasrifu say-yi-a-ha illa ant)
My suggestions to this matter would be to utter this Du'a from time to time, and Allah knows best.