It isn't just about a preference between those different ways of saying the same thing. Remember that "singing" can be both a noun for the activity of singing as well as the present continuous form of the verb to sing. So, when someone says "I like her singing" it could mean that they like it when she sings, or that they like the singing that she does.
You are far more likely to hear someone say about their favourite singing artiste that they like their singing, because they are stating a preference for that artiste's style or vocal performance over others.
To say "I like it when [x] sings" suggests that [x] does other things as well besides singing, so one is more likely to say this about someone they know such as their child, or perhaps about someone who is known for other kinds of performing as well.