Wa alaykum a'salam. Good question. Allah said: "Every deed of the son of Adam is for him except fasting; it is for Me and I shall reward for it..." Al-Bukhari (1761) and Muslim (1946)
We are allowed to fast on any days except for a few which are: the two days of Eid, and the three days of Ayyam al Tashreeq (these are the days following Eid al Adha). And we can fast on Friday, but we cannot single out every Friday to fast on.
There is also a Hadith (Hasan): "Do not fast on Saturdays apart from days when you are obliged to fast. If any one of you cannot find anything other than grape stalks or the twigs of a tree, let him chew it (to make sure that he is not fasting)." from al-Tirmidhi (744), Abu Dawood (2421) and Ibn Maajah (1726) - It just means we also can't single out only Saturday to fast, unless we are doing Qada or it is Ayyam al Beed, etc.
I don't know of any other prohibitions except for those above..
It is Sunnah to fast on Monday and Thursday because this is the day our deeds are shown to Allah.
RasoolAllah salaAllahu alayhi wa salam also advised us to fast three days each month, they can be any three days but preferably the 13th, 14th, and 15th (of the Islamic calendar) because these are Al Ayyam al Beed (the white days) when the moon is full.
There are also other days, we see from Ahadeeth, which Nabi salaAllahu alayhi wa salam prescribed for us to fast:
Ashoora (10th Muharram)
Arafah (9th Dhul Hijjah)
6 days in the month of Shawwal
In the month of Muharram and the month of Sha'baan, whenever we can
And this is all supported by the Sunnah, it is always best to check sources, I'm sorry I didn't provide a source for them all. I don't know if I missed something, in shaa Allah someone else can correct me if I did.
Allah knows best.
