Many have heard of the prohibition in Judaism for a man to hear a woman sing, called kol isha. The source for this is in the Talmud:
Shemuel said: The voice of a woman is nakedness (kol b’isha erva) as it says (Song of Songs 2:14) ‘for your voice is sweet and your countenance comely.’
(Talmud Berakhot 24a)
There are a few questions associated with this prohibition. What is the scope of the prohibition? Is it always prohibited or only sometimes? What exactly counts as the "voice" of a woman?
The topic is broad, but the general consensus in orthodoxy is that the prohibition applies at all times (even though it is stated in the Talmud in the context of praying the Shma), but it is only a woman’s singing that is prohibited, not speaking in general. If rationale behind the prohibition is that a woman’s voice will lead men to sin, it makes sense to allow normal talk, since all men are used to hearing women talk during normal life.
So what about singing? Some people say any singing even of girls is prohibited, but there are several possible leniencies in situations where it is unlikely that listening to a woman sing will lead a man to sin. Many people allow the voice of family members. Some authorities say that it is permitted if the man does not know what the woman looks like, e.g., on the radio. Another case that some authorities say is ok, is when the content of the song is definitely not sexual in any way, e.g., a shabat song, funeral song or lullaby. And finally, allowances can be made if there is a group of women singing together as "two voices cannot be distinguished".
What is clear in any case is, that the prohibition only applies to men. Women may hear other women sing without problems. There are actual all-women-bands who perform only for women audiences, e.g., the Bulletproof Stockings. So if a woman has talent as a singer, there are ways she can develop this talent.