Making du`aa’ (supplication) after the last tashahhud in prayer and before the tasleem is permissible and recommended without any restriction. It is optional to select the supplication you prefer whether it is from the Sunnah or the Qur’an. In this regard, it was narrated that the Prophet (peace be upon him) said that after reciting the tashahhud, “Then select the invocation you like best and recite it.” (Al-Bukhari)
However there are some supplications which are quoted in hadiths and which should be taken in first place. Note that none of these invocations are from the Qur’an, but this for sure does not mean we should avoid supplications in the Qur’an.
Abu Hurairah (may Allah be pleased with him) narrated that the Messenger of Allah (peace be upon him) said: When any one of you utters tashahhud (in prayer) he must seek refuge with Allah from four (trials) and should thus say:” O Allah! I seek refuge with You from the torment of the Hell, from the torment of the grave, from the trial of life and death and from the evil of the trial of Masih Ad-Dajjal (Antichrist).” (Muslim)
In another hadith, `A’ishah, the wife of the Messenger of Allah (peace be upon him) narrated, that the Messenger of Allah (peace be upon him) used to supplicate in prayer thus: “O Allah! I seek refuge with You from the torment of the grave, and I seek refuge with You from the trial of the Masih Ad-Dajjal (Antichrist) and I seek refuge with You from the trial of life and death. O Allah! I seek refuge with You from sin and debt.” She (`A’ishah) reported: Someone said to him – (the Holy Prophet): Messenger of Allah! why is it that you so often seek refuge from debt? He said: When a (person) incurs debt, (he is obliged) to tell lies and break promise.” (Muslim)
For more information and rulings regarding making supplication after tashahhud, watch this short talk with Dr. Muhammad Salah.
Source: Huda Youtube Channel.