Fasting in the month of Shábaan, without the stipulation of any day, is proven from Rasulullah sallallahu alayhi wa sallam. In fact, he used to fast excessively during this month. A Hadith of Hazrat Aisha (radiyallahu anha) wherein she describes the fast of Rasulullah (sallallahu alayhi wa sallam) substantiates this. She says that aside from the month of Ramadaan, he never used to fast for an entire month. However, of the remaining months, he would fast in Shábaan the most.
Bukhari, vol. 1, pg. 264

Fast of the 15th Shábaan

On the day immediately following the Night of Bara'at, i.e. the 15th of Shábaan, it is advisable to fast.

Though majority of the scholars of Hadith have certain doubts regarding the authenticity of a narration in Ibn Majah, advising one to fast on this day, stating it is extremely weak, it has been mentioned earlier that the fasts of the first half of Shábaan have special merits and Rasulullah (sallallahu alayhi wa sallam) used to fast most of the days in Shábaan. Moreover, a large number of the elders (salaf) of the Ummah have been observing the fast of the 15th Shábaan. This constant practice of our learned and pious predecessors establishes its Istihbaab (preference).

Therefore, it is advisable to fast the 15th of Shábaan as an optional (Nafl) fast. One can also keep a fast of Qada' on this day and it is hoped that he can also benefit himself from the merits of this fast.

NB: Fasting on the 13th, 14th and 15th of the Islamic months is an established Sunnah practice.

And Allah Ta'ala Knows Best

Was Salaam

Shafique Jakhura (Mufti) MD Mangera (Mufti)