Does Halacha allow taking a haircut on one of the public fast days, such as Shiba Asar Be’Tammuz, Som Gedalya or Asara Be’Tebet? (It goes without saying that haircutting is forbidden on Yom Kippur, when all the restrictions of Shabbat apply, and on Tisha B’Ab.)
The Eliyahu Rabba (Rav Eliyahu Shapiro of Prague, 1660-1712) established a rule (in Siman 551) that anything forbidden by Halacha during the nine days from Rosh Hodesh Ab through Tisha B’Ab is also forbidden on a fast day. According to this position, it would be forbidden for Ashkenazim to take a haircut on a fast day, because Ashkenazim follow the custom of refraining from haircutting throughout the three weeks from Shiba Asar Be’Tammuz until Tisha B’Ab. For them, according to the Eliyahu Rabba, haircutting would be forbidden on fast days just as it is forbidden during the Nine Days. Sepharadim, however, do not forbid haircuts during the Nine Days; our custom is to forbid haircutting only during “Shabu’a She’hal Bo” – the week of Tisha B’Ab. According to the custom of the Sepharadim, then, haircutting would be permissible on fast days, even if one accepts the theory of the Eliyahu Rabba.
Interestingly enough, Rav Eliezer Waldenberg (Israel, 1915-2006), in his work Sitz Eliezer, claims that the Eliyahu Rabba refers only to prohibitions that relate to bodily enjoyment. Haircutting, of course, is not forbidden because of any kind of physical enjoyment it brings. Therefore, the Sitz Eliezer argues, even Ashkenazim would allow taking haircuts on fast days, as haircutting is not included in the Eliyahu Rabba’s rule. Of course, Ashkenazim may not take a haircut on Shiba Asar Be’Tammuz, as they forbid haircutting throughout the three-week period from Shiba Asar Be’Tammuz through Tisha B’Ab.
Some other authorities, however, rule that haircutting is indeed forbidden on fast days, for both Ashkenazim and Sepharadim. Rav Haim Palachi (Turkey, 1788-1869), in his work Ru’ah Haim, arrives at this conclusion on the basis of a story told in the Gemara, in Masechet Rosh Hashanah. Once the Rabbis of Lod declared a fast day due to a drought, and they later realized that the day they had declared as a fast day was during the holiday of Hanukah, when fasting is forbidden. In order to demonstrate that a fast would not be observed on the assigned day, Rabbi Yehoshua took a haircut. The Tureh Eben (Rav Aryeh Leib Ginzburg, 1695-1785) notes that if haircutting served as an indication that the day was not a fast, then quite obviously haircutting is forbidden on fast days. This is the ruling of Rav Haim Palachi, as mentioned, and also of Hacham Ben Sion Abba Shaul (Israel, 1923-1998), in his work Or Le’sion.
Hacham Ovadia Yosef, however, ruled leniently in this regard, and allowed haircutting on fast days. In his view, there is no reason to refrain from haircutting on a fast day even as a measure of extra piety.
As for the final Halacha, then, it is permissible to take a haircut on a fast day, though if one wishes to follow the stringent view and refrain from haircutting on fast days, then “Tabo Alav Beracha” (he is worthy of blessing).
Summary: It is permissible to take a haircut on a fast day (except, of course, on Tisha B’Ab; Ashkenazim also refrain from haircutting from Shiba Asar Be’Tammuz). Some authorities, however, are stringent in this regard.