Yelle

Edith Piaf

Poni Hoax

Alif Tree
LornaLee
Daft Punk lol.

Poni Hoax – Budapest
This is a bit old, though.

Are you looking for them to sing in English?

There is a French singer named Soko, but she doesn’t sing in French. She’s awesome.

Not to sing. Just listen to. But should be singing in french

Juliette Gre’co. I love Sous le Ciel de Paris.

@zenvelo I’ve heard the Edith Piaff rendition. Beautiful

If you are looking for the traditional French singers, here’s a representative list.

Jacques Brel (sings also in Dutch), Yves Montand. Patachou, Georges Brassens, Juliette Greco, Yvette Guilbert. Piaf (yawn). I have wonderful old vinyls of them singing.
Here’s a list of more traditional, regional and pure French singers.
http://en.wikipedia.org/wiki/French_folk_music

I have no knowledge of more contemporary music.

Toto? Not sure about the French part, but at one time they had a French singer.

Yael Naïm
You might recognize her most famous song, New Soul, which was featured in an Apple commercial. I have never heard her sing in French but I believe that she has songs in French and Hebrew, in addition to her English songs.

I like Telephone. They’re 70s/80s. A more recent band is Eiffel, who were very much influenced by the American band The Pixies.

You can also listen to FIP via online streaming. It’s a French national radio station, all music, that plays a very eclectic selection of French (and other) music, all styles and all time periods. It’s a great way to discover new music.

Stereo Total
Autour de Lucie

And another vote for Piaf.

Serge Gainsbourg is an interesting cat. A few songs:
La Javanaise Ne dis rien La Horse

Phoenix

Françoise Hardy.