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 is His slave and Messenger.
The actions of people are presented to Allah in the month of Sha’baan so the Prophet used to fast repeatedly during this month. ‘Aa’ishah 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 other than he did in Sha’baan." [Reported by al-Bukhari and Muslim]
Usamah ibn Zaid narrated "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’baan." The Prophet 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 and an-Nasaa’i]
As for fasting particularly on the fifteenth of Sha’baan, then this is not proved through any sound evidence.
Nonetheless, there are some sayings of the companions of the Prophet and their immediate followers which indicate the virtue of this night. But this does not mean that there is any special significance to that day or that night. So, there is no evidence proving that fasting such a day is Sunnah nor is there any evidence proving that there are some specific prayers to be performed on the night of the 15th of Sha’baan.
Allah knows best.