Here is an excellent lecture that I found on the internet by Hamzah Abdur-Razzaq (hafidhahullah Ta'ala) with regards to the subject matter of singling out Saturdays specifically for fasting. A very detailed lecture in this subject matter containing many benefits.
With regard to this matter of fasting on a Saturday that there could be several scenarios:
1– It may be that obligatory, such as observing a Ramadan fast during Ramadan or making it up later on, or observing a fast as expiation (kaffarah), or instead of offering a sacrifice when performing Hajj tamattu’, and so on.
2– If he fasts on the Friday before it.
3– If it happens to be one of the days when it is prescribed to fast, such as Ayyam al-Beed (the 13th, 14th and 15th of each Hijri month), ‘Arafah, ‘Ashura, six days of Shawwal for one who has fasted Ramadan, and the ninth of Dhul-Hijjah.
3– If it happens to be one of the days when it is prescribed to fast, such as Ayyam al-Beed (the 13th, 14th and 15th of each Hijri month), ‘Arafah, ‘Ashura, six days of Shawwal for one who has fasted Ramadan, and the ninth of Dhul-Hijjah.
4– When it happens to be a day when he habitually fasts, such as a person whose habit is to fast alternate days, and the day that he fasts happens to be a Saturday.
5– If he singles it out to observe a voluntary fast for one day only. This is what is forbidden.
5– If he singles it out to observe a voluntary fast for one day only. This is what is forbidden.
May Allah enable us to fully benefit from that which we hear, Aameen.
Abu Hurairah (رضي الله عنه) narrated that the Prophet ﷺ said: "Whoever invites to guidance will be rewarded like those who follow him in the guidance are rewarded, and that will not decrease the reward of those who have followed him." [Collected by Imām Muslim]
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