The Anshei Kehillos Ashkenaz (Yekkes) accepted on themselves the added stringency of not even singing during the three weeks. To the exclusion of Shabbos and in order to enhance davening and learning.
As for everyone else, there is no specific Issur that I know of. However, if you’re going to sing, you may as well sing about Yerushalayim, Teshuva, etc…
Music in line with the spirit of the period would be appropriate: things like Eli Eli, songs based on mournful tehillim (the Mi Ma’akim perakim), verses from Eicha, etc. seem appropriate. One of the most haunting things I find myself listening to at this time is Antoine Brumel’s motet (a capella SATB), which is Eicha 1:8 and Eicha 1:11. Very moving.
I sing as I listen. As long as there are no instruments involved, and the music leads one to a state of introspection, what could the issue be? Does a person not sing when they go to shul? When the Rambam paskened that people shouldn’t play music in a world without a Beit haMikdash, he wasn’t referring to singing, but to actual zimrah.
R’ Moshe and the Tzitz Eliezer both say that one shouldn’t, but I think it’s clear that the Minhag is not like them, except for certain sections of the Yekke community.
As a baal tefillah, I am interested in this, Sam2. In Ashkenaz, did they change the nusach at all for the 3 weeks, so that certain emotions wouldn’t be expressed?
sam4: They’re in the famous T’shuvos. R’ Moshe in OC 1:167 (I think; he says that Zimra D’mana is Assur and that during the 3 weeks and S’firah D’puma would also be Assur). The TZ”E quotes him and agrees in 15:33.