During the three weeks (of mourning for the destruction of the Temple, between the 17th of Tammuz and the 9th of Av) can I get a haircut on erev Shabbos (the eve of Sabbath) or on Rosh Chodesh (the start of the new month) [i.e., what is the custom/minhag during this period]?
The accepted custom is to not cut any hair on the body during the entire three week period, period. This includes Shabbat and Rosh Hodesh. Were it otherwise, it really would not ever be a three-week custom, since everyone would get a haircut and/or shave for Shabbat or Rosh Hodesh.
There is allowance for trimming the moustache prior to the week in which Tish'ah B'Av occurs, if it interferes with eating.
One of the outward signs of mourning is disheveled hair. Those grieving a personal loss will abstain from cutting their hair during the appropriate mourning period. The same is true for grieving a national loss. And there is no deeper a national loss than the destruction of the Jerusalem Temples. Thus the regnant custom among European Jews and their descendants is to abstain from cutting their hair during the entire three-week period leading up to and including the Ninth of Av (cf. Rabbi Moses Isserles, Shulhan Arukh, OrahHayyim 551:4). Subsequent authorities have allowed for a number of exceptions to the custom. First, a moustache may be trimmed should it interfere with eating. Second, one who regularly shaves may shave before Shabbat. Shaving, however, differs from haircutting. Third, haircutting is allowed if disheveled hair would result in a loss of business customers and certainly a loss of a job. Fourth, the principals (that is, father of the baby, mohel and sandek) may cut their hair in preparation for the Brit Milah. Other exceptions might be warranted but only after consulting with a rabbi.
I note that your question is specifically NOT what does Halachah/Jewish Law say about this matter. You ask about minhag/custom.
Answering you from that perspective, I would point out that minhagim/customs differ in various communities. Clearly, in a very devout or observant community, the custom would follow the Halachah, and few or no persons would get a haircut as you describe the situation. On the other hand, in a community where the bulk of the members were Reform, this would be a matter of personal decision, and some might refrain, while others (perhaps the majority) would have the haricut.
The question you ask presupposes a universal response, and that just isn't likely.
The situation is not unlike the concern about naming a child: in the Ashkenazic Jewish community, it would be considered against the prevailing custom/minhag to name after a living relative, while in the Sephardic Jewish community, it would be considered in keeping with the prevailing custom/minhag to name after a living relative. Minhagim/customs differ from community to community, place to place, and tme to time.
If you are asking whether you should go against a prevailing minhag/custom, I would suggest that the presumption should be no, you should not violate minhag/custom, at least not without a very good reason. This same idea is expressed well in the English adage, 'When in Rome, do as the Romans.' That works, insofar as the 'Romans' we are talking about your those in the community you come from.
I hope that this has helped to answer your concern.
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