This is a point that directors frequently neglect; singing is hard work and if an actor is to look successful miming then they must be working hard. Glenn Close took singing lessons before recording her film in which she played an opera singer; I've not seen it so I can't comment about how successful she was. In an Inspector Morse episode, Frances Barber played an opera singer and made a complete hash of trying to pretend she was singing an aria (from Tosca I think). Whereas in the TV adaption of one of Jilly Cooper's novels, they employed Rosalind Plowright to play an opera singer with spectacularly successful results (the series also had Alison Moyet playing a pop diva).
I know this seems a small point, but each time I see someone on Film or TV purportedly singing, but actually completely failing, I get annoyed; given the amounts of money these things cost, surely they can get a little thing like singing right.