Is there a problem in halachah of listening to women singing?
It is not permitted to listen to a woman singing if the performance is “live.”
For recorded voice, there is room for leniency.
The Rambam (Issurei Biah end of Chap. 21), the Rosh (siman 37) and the Tur (Even Ha-Ezer 21) write that it is forbidden to hear the voice of women singing. Their rulings imply that this prohibition applies at all times, and not specifically while reciting the Shema (which the Gemara refers to; see Berachos 24).
The prohibition is likewise recorded by the Shulchan Aruch (21:1), and see also Shut Chasam Sofer (Choshen Mishpat 190). Based on these clear rulings, it is not permitted, even not during Kerias Shema or davening (or Torah study) to listen to women’s singing. This ruling is explicitly given by the Mishnah Berurah (75:17).
However, a number of authorities write that this applies specifically to a woman who is known to the listener, and that the prohibition therefore doesn’t apply to recorded singing (see Yabia Omer, Vol. 9, Orach Chaim 108, sec. 43; see also Yabia Omer 1:6; Halachah Berurah 75:15; Benei Banim 2:11; 3:127).
The Tzitz Eliezer (5:2) writes that in addition, recorded singing is not considered an actual voice, and therefore the prohibition doesn’t apply (see also Beis She’arim, no. 33).
Note that other authorities are stringent: see Peri Hasadeh (3:35); Shut Chelkas Yaakov (Orach Chaim 6); Shut Ravaz (teshuvos from author’s son, no. 6); Shevet Halevi (3:81); Az Nidberu (9:9). See also Chayei Asher (104), who mentions both sides of the issue.
Because this is not a question of a Torah prohibition, there is room for leniency in practice.