All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad ﷺ is His slave and Messenger.
The actions of people are presented to Allah in the month of Sha’ban, so the Prophet ﷺ used to fast repeatedly in this month. ‘Aa’ishah, may Allah be pleased with her, said: "I never saw the Messenger of Allah fast for a complete month except for Ramadan, and I never saw him do more fasting in any month than he did in Sha’ban". [Al-Bukhari and Muslim]
Usamah bin Zaid, may Allah be pleased with him, reported "I said to the Prophet, O Messenger of Allah! I did not see you fasting in any month as you do in the month of Sha’ban". The Prophet ﷺ said: "People neglect this month which is between Rajab and Ramadan, in this month the actions of the people are presented to Allah; so I like my deeds to be presented while I am fasting". [Abu Dawood, An-Nassaa’i]
As for fasting particularly the fifteenth of Sha’ban this is not proved through any sound evidence. There are some sayings of the companions of the Prophet and their successors, may Allah be pleased with them, which indicate the virtue of this night. But this does not mean that there is any special significance to this particular day or this specific night.
So, there is no evidence proving that fasting such a day is a Sunnah nor is there any evidence proving that there are some specific prayers to be performed in the fifteenth night of Sha’ban.