We should not fast only on a Friday. The Holy Prophet sas said:
None of you should fast on Friday unless he fasts a day before or after it.
(Sahih al-Bukhari, Book 30, Hadith 92; Sahih Muslim, Book 13, Hadith 189).
There is some disagreement about fasting only on Saturday. It is narrated that the Holy Prophet sas said:
Do not fast on Saturday except for what has been made obligatory upon you (by Allah).
(Sunan at-Tirmadhi
, vol. 2, Book 3, Hadith 744; Sunan Ibn Majah, vol. 1, Book 7, Hadith 1,726)
Imam Malik considered this narration false (Sunan Abi Dawud, Book 13, Hadith 2,418), and Abu Dawud considered it abrogated, however, it was otherwise considered reliable (Bulugh-al-Maram, Book 5, Hadith 712). According to Fiqh Ahmadiyya, under disapproved and disliked days for fasting, choosing Friday only and Saturday only are mentioned (Fiqh Ahmadiyya, vol. 1, p. 272). It would be best to not fast only on Saturday.