The free-jazz / improv band The Recyclers (ie Benoît Delbecq on piano and prepared piano, Steve Argüelles on drums and percussions and Noël Akchoté on guitar, bass, etc.) have a great cover album titled Morceaux Choisis, published by Rectangle in 1997. Joined by singers Katerine, Ignatus, Sasha Andrès and Irène Jacob, they play staples of "chanson française", old (Francis Poulenc) and new-ish (Polyphonic Size, a Belgian "new-wave" band from the 80s). Songwriters covered include Georges Moustaki, Brigitte Fontaine, Serge Gainsbourg, Léo Ferré, Jacques Brel, Nino Ferrer and Pierre Barouh (founder of the Saravah label that's a gold mine for quirky 70s music). Great stuff there.
Here's Brigitte Fontaine "La vache enragée" as sung by Katerine :
Brigitte Fontaine one of the living greats, but her stuff is probably too odd for someone not experienced in the language. Among the classics, I'd say Jacques Brel's the apex. Check out his live stuff from youtube to see what it was all about.