On which days of the year is it forbidden to fast a private fast?

Good day,

It is forbidden to fast on Shabbat (Shulchan Aruch, siman 288, se'if 1), Yom Tov (see there, siman 611, se'if 1), Rosh Chodesh (there, siman 417, se'if 1), Chol HaMoed (there, siman 547, se'if 1), Erev Yom Kippur (Rema, siman 604, se'if 1), Chanukah (Shulchan Aruch, siman 670, se'if 1. And regarding Erev Chanukah, see Shulchan Aruch, siman 686, se'if 1, and Mishnah Berurah there), 15 of Shevat (Magen Avraham, siman 673, sk"a 1), Purim (Shulchan Aruch, siman 696, se'if 3), and 15 of Av (Magen Avraham, siman 673 there. And see also Shulchan Aruch, siman 678, and Mishnah Berurah there).

It is also customary not to fast on Lag BaOmer, Rosh Chodesh Sivan until Shavuot, and on the days between Yom Kippur and Sukkot (Magen Avraham, siman 673 there). 

According to Sephardi custom, it is permitted to fast a private fast in the month of Nisan, except for Rosh Chodesh Nisan and the seven days of Pesach (Shulchan Aruch, siman 429, se'if 2, and Chazon Ovadia, Pesach, part 2, p. 5), while the custom of Ashkenazim is to refrain from this (Rema there).

It is forbidden to fast on Isru Chag of Shavuot (Shulchan Aruch, siman 494, se'if 3), and on other Isru Chag days, according to Sephardi custom it is permitted, though it is good to be stringent (Chazon Ovadia there), while the custom of Ashkenazim is to refrain from this (Mishnah Berurah, siman 429, se"k 14).

The halachic authorities differ regarding whether it is permitted to fast on Pesach Sheni, as explained in the sources.

Sources: In Eshel Avraham (siman 131, ד"ה הנוהגין) it is written that if the בה"ב fast falls on Pesach Sheni, one may fast. Likewise, this is written in Responsa Zikhron Yehudah, part 1 (siman 166), and in Responsa Maharsham, part 6 (siman 32). See also his work Da'at Torah (end of siman 492), where he brought from Sefer Machberet HaKodesh by the Ram"a regarding the intent of the counting, which states that one should not fast on Pesach Sheni, but in a year in which בה"ב falls then, the public may fast on it.

However, the view of Rabbi Yosef Shaul Natanson z"l in his book Divrei Shaul - Yosef Da'at (siman 401) is that one should not fast on it. And in the above-mentioned Responsa Maharsham, in the gloss of the author's grandson Rabbi Shalom Shvadron z"l, it is brought that he heard from Rabbi Yaakov Moshe Charlap z"l that the leaders in the holy city of Jerusalem, may it be rebuilt and established, also discussed this, and they found written in the ledger of the synagogue of the Hurva of Rabbi Yehuda HeChassid z"l that if such a case occurs, they should not fast בה"ב, but only two fasts בלבד.

On Luach Eretz Yisrael it is written that among those who fast בה"ב, some do not fast at all on this day (and some fast the next day), and some fast only until midday. See also Responsa Mishneh Halachot, part 11 (siman 204), which writes that if one customarily fasts, it is permitted; nevertheless, one who refrains should refrain, in accordance with the ruling of the aforementioned elder, Yosef Da'at.

 With blessings,

Hillel Meirs