All perfect praise be to Allah, The Lord of The Worlds, and may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Cases of singling out Saturday for fasting are as follows:
First: Fasting on Saturday to make up for missed obligatory fast or to expiate an oath or to fulfill a vow or it (Saturday) coincided with the day of Ashura, the day of Arafah, the white days and other days recommended for fasting or it coincided with a habit like the one who fasts every other day: Scholars of the four legal schools have agreed that this type of fasting is recommended, even if it coincided with a Saturday.
Second: Singling out Saturday for offering absolute voluntary fast without coinciding with any occasion, as those mentioned in the former case, and without fasting a day before or after it. In this case, fasting is properly disapproved according to our Madhab (Shaffii school of thought) and that of the majority of the Muslim scholars. This view rests on the Hadith where the Prophet(PBUH) said: "Do not fast on Saturday except for what has been made obligatory upon you (by Allah). If one of you does not find but a grape peal or a tree's twig, then let him chew it." {Abu Dawood Hadith No. 2421}.
For further details, please refer to Al-Ramli`s [Nehayat al-Mohtajj pp.209/ vol.3]. And Allah Knows Best.