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…