We should all get out more.
‘I like this song. Who sings it?’ is both common and readily understandable. In the context Indranil described I’ve heard those very words recently, and also used them myself.
I’d generally agree with John about ‘Habitual present tense’ and ‘present tense of immediacy' being just descriptiions, but not in this case.
Whatever those labels meant to the British Council it seems obvious that if the choice is as given, then the example clearly refers to a habitual action with no sense of immediacy.
Almost equally clearly, ‘who sings it’ and ‘who’s singing it’ and ‘who sang it’ are not interchangeable and Indranil’s entitled to a bit of help, here.
Very likely, as Kate said, ‘who sings it’ means ‘which one artist or group is most associated with it?’
Very likely, ‘who’s singing it’ means ‘whose version are we listening to now?’
‘Who sang it’ might be much more tricky. It might mean ‘whose version were we listening to five minutes ago?’ It also might mean ‘who most famously used to sing it when it was popular, 20 years ago?’
Worse, ‘who’s singing it’ could be ‘whose version are we listening to just now?’ and ‘Who sang it’ could be ‘whose version were we listening to just now?’ with the similarity of the two different ‘just nows’ confusing the issue.