Can I fast on Friday?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Singling out Friday as a day for fasting and its night as a night to perform prayer is forbidden in Shariah.  However, fasting Friday is permissible in two cases.

First, if one fasts a day before Friday or a day after it for the Hadith of Abu Hurairah who narrated that the Prophet (Sallallahu Alaihi wa Sallam) said: “Do not fast on Friday unless you fast on it together with the day before or the day after.” [Reported by Imam Al Bukhari]

Second, one may fast on a Friday if it coincides with his habitual fast such as when it coincides with the Aashurah day (tenth of Muharram), in this case it is permissible for him to fast.  The Prophet (Sallallahu Alaihi wa Sallam) said: "Do not exclusively choose the night of Friday [Thursday night in English] as a special night for performing the night prayers. Also, do not exclusively choose Friday as a day of fasting unless it occurs on a day that you regularly fast." [Reported by Imam Muslim]

Therefore, it is dislikeable to single out Friday for fasting.  But if one fasts a day before it or after it or he takes an oath to fast on the day of getting cured from a disease and that day happens to be Friday, due to the above reasons fasting on that day is not Makrooh (dislikable). [Quoted from Imam Al Nawawi]

Allah knows best.