I only caught a part of "Little Voice" earlier this year, but what I remember was the actress who played the girl, Jane Horrocks. She did all of the singing in the film, no voice-overs. Other roles have included the completely ditzy assistant in "Absolutely Fabulous" and she has been a regular for the BBC for years.
Not arguing with your point about the film's themes, but Jane Horrocks is excellent, one of those talents who for some reason never seem to attract much attention outside the UK.
Actually, Horrocks plays two roles on AbFab, she is also the obnoxious Katie Grin. Yeah, she's a good singer, no question. I'm just telling my interpretation of the movie.