Should a woman avoid singing in front of her husband while she is a niddah? Is it strictly prohibited or just inadvisable? (Obviously I am not talking about deliberately seductive songs.)
Per Rabbi Shimon Eider's Sefer Hilchos Nidah one should "refrain from listening to his wife's singing when she is a Niddah."
R Yosef Shalom Elyashiv is quoted in Mishmeret HaTaharah (by his student R Moshe Mordechai Karp) Siman 195 footnote 207 and 209 as ruling that if her husband is used to hearing her voice (רגיל בו) and she isn't singing specifically to him (whereby R' Elyashiv thinks there would be a general problem of levity שחוק וקלות ראש) then there would be no prohibition on his hearing it and thus, seemingly, no prohibition on her singing.
Moreover, R' Karp notes in footnote 208 his own opinion that all the positions who prohibit her singing are only speaking of a case where he isn't used to hearing her voice (אינו רגיל בו). [I think he means all the older opinions such as the Pitchei Teshuva or Lechem veSimla, as there are some modern opinions about which it is hard to say this. --AA]
In Taharah Kehalocha (vol. 1 pg. 201), Rabbi Y. Farkash cites the Kitzur Shulchan Aruch (153:10) who forbids and the Pischei Teshuva (YD 195:10) and Igros Moshe (YD 2:75) who are in doubt about the matter but tend to be stringent. Accordingly, he rules that one should be stringent in the matter wherever a Kol Isha concern would arise with a non-relative1. The Minchas Yitzchok (7:70) additionally prohibits even musical instruments if she plays them for her husband's enjoyment.
However, there are those who disagree - claiming that listening to one's niddah wife singing cannot be compared to listening to other women. They claim that gazing at parts of a women's body that are usually uncovered is certainly worse than hearing her sing, and since this is permitted when his wife is niddah (Shulchan Aruch YD 195:7), it is certainly permitted to hear her sing (Suga Bashoshanim 195:2, and Mishnas Yaakov vol. 3 pg. 117 in the name of Chemdas Moshe (58)).
The Ben Ish Chai (Year 2, Parshas Tzav 25), after writing that the wife should not put the baby to sleep by singing to it when she is niddah if her husband can hear, writes that, "If the baby is crying alot and will not fall asleep any other way and there is nowhere for the husband to go - it seems that one could be lenient."
Based on the above, Rabbi Farkash concludes that although we should follow all those who are strict, in a time of difficulty where there is no other option one may be lenient.
1 His assessment of that includes his citing the Chok Yaakov (OC 479:6), Yosef Ometz (603) and Otzer Haposkim in the the name of other poskim who forbid listening to a woman singing Zemiros on Shabbos, as well as the Shevet Halevi (Shu"t 5:197:4 and Shiurim 6:2 "Vegam B'Zmiros") who applies such a prohibition even when there are other people singing along (see also Suga Bashoshanim 16:6-7 and Be'er Moshe there).