Albert Marcoeur

French avant-garde acoustic jazz, filled with moments of absurdity and cacophony. From his self-titled LP from 1974.
"Tu Tapes Trop Fort":

"Simone":

Lizzy Mercier Descloux

a French progenitor(tress?) of early no wave (writing as Rosa Yemen) and later various new wave, disco, and worldbeat sounds, Lizzy Mercier Descloux certainly did not stick to one style of songwriting. The following song, from her second album (as Lizzy), Mambo Nassau (1981) perhaps best melded her early no-wave and new wave roots with her danceable beats and definitely funk-ified world music consciousness:

Jackie Quartz

"Émotion (Le Jour Se Lève)" single, 1987:

Trisomie 21

Trisomie 21 has been a highly regarded staple in French electronic music since the early 80's up until the present day. "Memories" from Chapter IV (1986):

Brigitte Bardot Et Serge Gainsbourg

Brigitte Bardot, French singer/actress, teamed up with musician/actor/director Serge Gainsbourg to make several recordings in the late 60's. Gainsbourg's experimental, widely varied sound greatly influenced French popular and avant-garde music.
The duo originally recorded the "Bonnie and Clyde" single in 1967, and Gainsbourg released it in 1968:

Plaisir

french funkiness, "Visa Pour Aimer" (1984):

Martin Dupont

"It's So" from Sleep Is A Luxury (1985)- fan video:

Les Rita Mitsouko

"Don't Forget the Nite" from their 1982 single, Minuit Dansant: