Of the reported supplications in this regard is the Hadith that was reported by Abu Dawud, Ibn Majah, and others on the authority of Abu Salih from some of the Prophet's Companions that the Messenger of Allah (peace be upon him) said to a man: "What do you say in Salah? He replied: I first recite Tashah-hud (supplication recited in sitting position), and then I say: O Allah, I ask You for Paradise, and I seek refuge in You from Hell-Fire, but I do not understand your sound and the sound of Mu`adh (what you say or he says in Salah). The Prophet (peace be upon him) said: We too go around it (paradise and Hell-fire)." [Sunan Abu Dawud No. 792, and Sunan Ibn Majah No. 910, and was graded as authentic by Al Albany (may Allah bestow mercy on his soul) in Sahih Ibn Majah No. 742].
It was reported in the Sunnah some invocations that are to be said in Salah, however their positions are not known. It is better to recite these invocations in two positions: Either in prostration or after Tashah-hud because the Sunnah stressed supplication in these two positions.
Of these invocations is the Hadith that was reported in Sahih Al Bukhari and Sahih Muslim on the authority of Abu Bakr As-Siddiq (may Allah be pleased with him) that the Prophet (peace be upon him) said: "Teach me a good invocation that I use in Salah. He told me to say: Allahumma inny Zalumtu Nafsi Zulman Kathiran. Wala Yaghfirudh-dhunuba Illa Anta Faghfirli Maghfiratan Min `Indika War-Hamni Innaka Antal-Ghafurur-Rahim (O Allah! I have done great injustice to myself and none except You forgives sins, so please forgive me and be Merciful to me as You are the Forgiver, the Merciful)." [Sahih Al Bukhari No. 834, and Sahih Muslim No. 2705].
There is also the Hadith that was reported by An-Nasa'y on the authority of `Ata' ibn As-Sa'ib from his father (may Allah be pleased with him) who said: "`Ammar ibn Yasir led us in Salah and he made it brief. Some of the people said to him: 'You made the Salah short (or brief).' He said: 'Nevertheless, I still recited supplications that I heard from the Messenger of Allah (peace be upon him).' When he got up and left, a man -he was my father but he did not name himself- followed him and asked him about that supplication, then he came and told the people. "Allahumma bi `Ilmikal-Ghaiba wa Qudratika `Alal-Khalqi Ahyiny Ma `Alimtal-Hayata Khairan li, Wa Tawaffany Idha `Alimtal-Wafata Khayran li. Allahumma As'aluka Khashyataka Fil-Ghaybi wash-shahadat, wa As'aluka Kalimatul-Haqqi fir-Rida wal Ghadab. Wa As'alukal-Qasda fil Faqr wal-Ghina, wa As'aluka Na`iman la Yanfadu wa As'aluka Qurrata `Aynin la Tanqati`u wa As'alukar-Rida Ba`dal-Qada' wa As'aluka Bardal `Ayshi Ba`dal-Mawt, wa As'aluka Ladh-dhatan-Nazari ila wajhika Wash-shawqa ila Liqa'ika fi Ghayri Darra' Mudirrah wala Fitnatin Mudillah. Allahumma Zayyina Bizinatil-Iman, waj`alna Hudatan Muhtadin (O Alah, 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. O 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 at 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. O Allah, beautify us with the adornment of faith and make us among those who guide and are rightly guided." [Sunan An-Nasa'y No. 1305, and Al Albany (may Allah bestow mercy on his soul) graded it as authentic in Sahih Al Jami` No. 1301]. It is a great Hadith that was authentically reported from the Prophet (peace be upon him) and includes great benefits, noble objectives, and blessed ends.
Al Hafizh Ibn Rajab (may Allah bestow mercy on his soul) wrote a nice letter in the explanation of this Hadith and showing its meanings. Moreover, it is a useful letter, and perhaps I shall pause with some references of this Hadith and its great meanings to be helpful to us —by the will of Allah— to fulfill its meanings, May Allah grant us success!