French:
Camille. She's probably the best vocalist in the world. You might have to hear her stuff to know what I mean. Just a brilliant Avant Garde cool french chick.
Anaïs is a fantastic singer/songwriter who's live stuff makes me really wish I could speak French fluently.
Nouvelle Vague are an effortlessly hip and chic French group who use guest singers to do bossa nova covers of 80s new wave/alternative songs.
Hollywood Mon Amour is the sister project of Nouvelle Vague who do covers of movie songs (also generally from the 80s).
-M- (Mathieu Chedid) is another wonderful French songwriter.

Of course this all contemporary French music. You should also obviously check out Francoise Hardy, Edith Piaf and Jacques Brel (the album Scott Walker Sings Jacques Brel is a wonderful English introduction to his music).

I've never been much of a Serge Gainsbourg fan but he does have a few great songs.

This is my favourite French song ever.