Sheikh Salman al-Oadah
You should not intentionally single out Friday for fasting, but if you observe a fast on Friday in additional to the day before it or the day after it, then it is alright. It is permissible for you to fast Thursday and Friday or Friday and Saturday.
The Prophet (peace be upon him) said: “One of you should not fast on Friday without also fasting either the day before it or the day after it.” [Sahîh al-Bukhârî (1985) and Sahîh Muslim (1144)]
The prohibition of fasting on Fridays alone is a general prohibition which has specific exceptions to it. One of these exceptions is that it is permissible to observe an obligatory fast on Friday without having to observe a fast on the day before it or the day after it.
Another exception is that you may fast on a Friday by itself if it just happens to fall on the day of an established Sunnah fast or happens to coincide with a day that is part of your regular fasting program, for instance, if you fast the 15th of every month or if you fast every other day.
The proof for this is where the Prophet (peace be upon him) said: “Do not single out the evening before Friday for offering the night prayer. And do not single out the day of Friday for fasting, except if a fast that you fast happens to fall on that day.” [Sahîh Muslim (1144)]
This includes the day of `Ashûrâ', since fasting on this day is strongly recommended by the Sunnah. It also includes fasting on the Day of `Arafah or with the intention of observing one of the six days of Shawwâl.
The hadîth prohibiting fasting on Saturday reads: "Do not fast on Saturday except for a fast that has been made obligatory upon you. If you find nothing other than the bast of a grape or the branch of a tree, then chew on that." [Sunan Abî Dâwûd (2421), Sunan al-Tirmidhî (744) and Sunan Ibn Mâjah (1726)]
However, this hadîth is inauthentic. It's chain of transmission is inconsistently narrated and its text is false.
Moreover, we have already seen the hadîth that instructs us to fast on Friday and "the day after it". This clearly shows us that Saturday is not a day wherein fasting is prohibited.
And Allah knows best.
Source: Islam Today