Regarding fasting on a Friday, Shaykh Ibn 'Uthaymeen said:
"But if Friday coincided with the day of ‘Arafah for example, and a person then fasted it [i.e. only the Friday], then there is no harm or objection [in doing so] because this man [questioner] fasted [the Friday] because of it being the day of ‘Arafah and not because it was merely a Friday [and] if Friday coincided with the day of ‘Aashooraa’ and he fasted it then there is no objection in him to singularly fast that that day since he fasted it because it is the day of ‘Aashooraa’ and not because it is merely a Friday."
The problem is resolved, with regards to this kind of situation, where the day of A'ashuraa falls on a Saturday. The Saturday of which the Messenger of Allah
has said in an authentic hadeeth:
"Do not fast on the day of Saturday except that which has been made obligated upon you,and if one of you does not find except a bark of a tree then let him chew it." In another wording:"let him swallow it."
The point of evidence being that the Messenger of Allah
prohibited fasting on a Saturday-an absolute prohibition- except what is obligated, like the fast of Ramadaan or for the fulfilment of a vow, if one vowed to fast a complete month or a week.
As for what is not obligated, like the fast of A'ashuraa or the fast of A'rafah, where they may coincide with a Saturday. Does one fast? No, one does not fast, as the authentic hadeeth mentions: "except that which is obligated upon you."