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":

Suicide

Alan Vega and Martin Rev from NYC, their first album from 1977, Suicide.
"Rocket USA":

"Cheree":

United States of America

Formed by electronic composer/musician Joseph Byrd at UCLA in 1967, United States of America artfully blended psychedelic sounds with chamber music, electronic experimentation, and avant-garde rock n roll. Their self titled LP (1968) was their only release.
The two songs I've selected showcase their soft rock meanderings mixed with mesmerizing tape and vocal effects and a calmer touch of psychedelia.
"Love Song for the Dead Che"

"Cloud Song"

Thomas Leer

Thomas Leer's early work masterfully blended blithe pop instrumentation with self-built synthesizers and experimental tape effects.
B-side on the Private Plane/International (1978) single, "International":

Tomo Akikawabaya

Japanese mystery man Tomo Akikawabaya made his brief appearance in the mid 80's with several self-released singles, each featuring model Rena Anju on the cover, and promptly fell off the face of the Earth. there are no pictures of this man known to exist, he gave no interviews, never publicly performed, and he has (presumably) not released any more music since 1986. All instruments and noises played by Tomo Akikawabaya.
"Diamond" from the 1985 single:

Side B - "Dream of No Pillow":

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:

Mark Lane

recorded in 1981, "Quest" from Kill Joy (2000) a collection of rare or unreleased recordings:

Cleaners from Venus

Cleaners from Venus; Martin Newell and Lol, "Only A Shadow" from Midnight Cleaners (1982):

Crash Course In Science

from Philadelphia, PA.
"Factory Forehead" from Signals From Pier 13 (1981):

Solid Space

another DIY tape release from early 80's UK, Solid Space with the album Space Museum (1982):
"Radio France"

"Destination Moon"

"Contemplation"

Pas de Deux

"Cardiocleptomanie" from Axe Ends (1983):

Nico

after Chelsea Girl, Nico recorded three more albums with John Cale, consisting mostly of original compositions written and perfomed by Nico on her harmonium with Cale providing the backing instrumentation.
"My Only Child" and "All That Is My Own" from Desertshore (1970) as they appear in Philippe Garrel's film, La cicatrice interieure:

"König" didn't make it onto Desertshore but was also in the film:

YU

YU is C.U. Squirm, Rod Miles, and Bently Shaft.
"Socks and Sacks":

"Not Only My Hand & Me":

"The Big Picture":

"Huxley's Tune":

Bernthøler

"Exterieur Nuit" (1982)

"Japanese Garden" (1982)

The Cocteau Twins

"Essence" from Four-Calendar Cafe (1993)
"Oomingmak" from the album Victorialand (1986)

Les Rita Mitsouko

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

Throbbing Gristle

"Walkabout", from the album 20 Jazz Funk Greats, recorded 1979 on Industrial records, re-released on Mute in 1983:

Non Band

"Duncan Dancin'" from their 1982 self-titled LP:

Grace Jones

"Demolition Man" from the album Nightclubbing (1981)