Al-jawab billahi at-taufeeq (the answer with Allah’s guidance)
Fasting On Rajab.
There is no Sunnah or Nafl fast for the month of Rajab specifically.
Hafiz ibn Rajab al-Hanbaliy [RA] states that there is no specific fasts for the month of Rajab established from Nabi [Sallallaahu Alayhi Wasallam] nor his companions [Sahaaba [Radhiallaahu Anhu].’ (Lataaiful Ma’arif pg.166; Cairo)
Fasting On Shabaan.
The tradition relating to the merits of fasting on 15th of Shabaan is, no doubt, a weak tradition, not competent to prove this practice to be a Sunnah or a formal Mustahabb, but it can be acted upon as a measure of precaution, provided that the practice is not taken as Sunnah or a formal Mustahabb. It is for these reasons that some Ulama and elders have been fasting on the 15th of Shabaan and have been taking it an advisable practice.
By Mufti Taqi Uthmani.
The Fast Of 15th Shaban (Revised opinion of Mufti Taqi Usmani), please click on this link to read more: http://www.central-mosque.com/fiqh/15fast.htm
Sha’ban: Merits, Do’s, and Dont’s (previous article of Mufti Taqi Uthmani)http://www.central-mosque.com/fiqh/shab1.htm
And Only Allah Ta’ala Knows Best.
Moulana Qamruz Zaman