It is a confirmed sunna in the final sitting, before the final salams, to make supplications (dua), after reciting the tashahhud (which is wajib), and sending blessings on the Prophet (which is a confirmed sunna).
It is mentioned in by Imam al-Tumurtashi in this Tanwir al-Absar:
“And [it is from the confirmed sunnas of the prayer] to supplicate”
Imam al-Haskafi clarified and conditioned this by saying, in his Durr al-Mukhtar, “(to supplicate) with that which is not possible to ask from humans.” [Durr al-Mukhtar, on the margins of Ibn Abidin’s Radd al-Muhtar, 1: 321]
Later in the text, he explains that,
“(and to supplicate) in Arabic, and it is not permitted (H: makruh tahriman) in another language – Nahr. [It is recommended to supplicate] for oneself, one’s believing parents, teachers, [H: and all believers].” [Durr, 1: 350-352]
One makes these supplications, as well as supplications in prostration (sujud) using that which is mentioned in the Qur’an and Sunnah or that which is not normally possible to ask other humans for, because the Messenger of Allah (Allah bless him and give him peace) said, “No human speech befits this prayer of ours.” (Related by Imam Muslim in his Sahih, from Sayyidina Ma`awiya (Allah be pleased with him))
Making supplications with that which is normally possible to ask humans for invalidates the prayer. So beware!
Therefore, instead of making specific worldly supplications, such as, “O Allah, marry me to Layla,” or, “O Allah, give me a red Honda Civic,” or, “Allah, I want mansaf tonight,” which would invalidate the prayer, one makes general supplications in Arabic, such as, “O Allah, I ask You of your bounty,” (Allahumma innee as’aluka min fadlik) with specific intentions in one’s heart.
If you don’t know Arabic, and want to supplicate for specific matters, then just say, “Ya Allah” or “Ya Rabb” repeatedly, intending the specific matters you are supplicating for.
In general, though, while there is no specific supplication that is a confirmed sunna, though if one recites one of the duas that are transmitted from the Beloved of Allah (upon him be blessings and peace), this is best.
And Allah knows best.