The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) rules that one should not take upon himself a voluntary Ta’anit Yahid (private fast) on a Friday. A person who wishes to accept upon himself a personal fast should not schedule the fast for Friday. The exception to this rule, the Ben Ish Hai writes, is fasts that one observes to commemorate a particular occasion. For example, many people have the custom to fast each year on the Yartzheit of a parent. If the Yartzheit falls on Friday, it is permissible to observe a fast on that day. Similarly, those who customarily fast on Ereb Rosh Hodesh (known as "Yom Kippur Katan") may observe the fast when Ereb Rosh Hodesh falls on Friday.
However, the Ben Ish Hai writes, when a person observes a fast on Friday, he should not fast until nightfall, as it is inappropriate to begin Shabbat in a state of discomfort. Therefore, when such an occasion falls on Friday, one should take it upon himself to fast only until midday or until some point before nightfall. Hacham Ovadia Yosef adds that in such a case one may recite Arbit early, before sundown, and then fast until sundown, at which point he should recite Kiddush and eat. In the summer months, it is very common for communities to recite Arbit and accept Shabbat early, before sunset, and thus this Halacha becomes quite relevant. A person who is fasting on a Friday during the summer to mark an occasion such as a Yartzheit or Ereb Rosh Hodesh may fast until sunset and then recite Kiddush. (In the winter months, of course, it would be very difficult, or nearly impossible, to find a Minyan accepting Shabbat early, so this option is not practical during the winter.)
The only time when fasting is allowed on Friday until nightfall is the fast of Asara Be’Tebet, which (unlike other public fast days) occasionally falls on Friday. This fast is a "Ta’anit Sibur," a public fast day observed by all Jews, and therefore it applies at a higher level of stringency, allowing us to fast until nightfall even on Friday.
Summary: Generally speaking, one should not observe a fast on Friday, except for the fast of Asara Ba’Tebet. One may observe a voluntary fast on Friday only for the purpose of marking a particular occasion, such as a Yartzheit or Ereb Rosh Hodesh. In such a case, however, one should not fast until nightfall, and should instead take it upon himself to fast until some point prior to nightfall.
