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":

Gale Garnett

Canadian-born singer Gale Garnett took a dip in the hippie pool around the end of the 60's with her psych-ish band Gentle Reign, but she started her career as a distinctly less-feminine female singer in the mid 50's. Here's her 1965 single, I'll Cry Alone with a wacky video for the loungy B-side, "Where Do You Go to Go Away?"
"I'll Cry Alone":

"Where Do You Go to Go Away?":

July

UK psychedelia from 1968, their self-titled LP. Where the rest of the album has a more predictable, albeit good, late 60's psych-pop sound, "The Way" sounds like a funky dub remix of a 60's song that could have been recorded yesterday, except that it is a 60's song, and it's actually not dubbed.
"The Way":

"I See":

Dream

Swedish psychedelic group Dream with their album Get Dreamy (1967).
"Do You Dream?"

Mikio Masuda

Japanese keyboardist Mikio Masuda with a legitimately smooth track, and quite possibly the best/worst album art I've seen in a while.
"Moon Stone" from the 1977 album of the same name:

Mort Garson

Mother Earth's Plantasia (1976) by Mort Garson, moog instrumentation for growing plants. Mort's career started as a composer and pianist in the early 60's, but he became known mostly for the eccentric psychedelia that doused his Moog recordings. Preceeding Plantasia was Black Mass (1971) written under the pseudonym of Lucifer and meant to accompany the ceremony of its namesake; a series of singles, one for each aspect of the Zodiac; and a psychedelic re-telling of the Wizard of Oz, The Wozard of Iz (1968), complete with insane narration.
"Symphony for a Spider Plant":

"Music to Soothe the Savage Snake Plant":

"Ode to an African Violet":

"Swingin' Spathipyllums":

Suicide

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

"Cheree":

Amon Düül II

Amon Düül II was one of the most influential krautrock conglomerations, with members constantly flowing in and out and branching into other groups and new musical territory. This particular period, around the release of Yeti (1970), turned out to be a seminal moment in the history of the group, marking the beginning of their underground success outside of Germany. The album shows a remarkable breadth of expression, from strange, trance-like moments of noise, constrained acoustic ragas and improvisations, to explosive, dirty, electric prog with slamming beats and tape effects.
"Archangel Thunderbird":

"Pale Gallery":

"Hallucination Guillotine":

Tyndall

Nik Tyndall and Rudolf Langer with their 3rd LP, Reflexionen (1982).
"Strassen Der Nacht":

Human League

the Human League from later in their career with their 1986 single, "Human":

Angelo Badalamenti

Angelo Badalamenti recorded the scores for several of David Lynch's films.
"Twin Peaks Theme", as it appears in the opening segment of the pilot episode of David Lynch's Twin Peaks:

"Fred's World" from the Lost Highway soundtrack (1997):

"Mysteries of Love" as it appears in Blue Velvet (1987):

Billy Idol

if the 1980's had just one douche bag to call it's own, it would be Billy Idol. here he is showing his less "hard" side with two of his most popular ballads.
"Eyes Without A Face" from Rebel Yell (1983):

"Sweet Sixteen" from Whiplash Smile (1986):

Unknown DJ

Andre Manuel, the Unknown DJ, early west-coast electro beats.
"Basstronic" single, 1988:

better audio: http://youtu.be/ytddqtWSP-E

Martin Newell

Martin Newell was a strong force for the DIY camp during the new wave days. He produced, recorded, and played (nearly) all of the instrumentation for his solo recordings. He also wrote and played under the name Cleaners from Venus with Lol Elliot in the early 80's, releasing most of their work together from home on cassette. The following are from his first cassette as Martin Newell, Songs for a Fallow Land (1985).
"Julie Profumo":

"Gamma Ray Blue":

Gandalf

american psych band, Gandalf with "Tiffany Rings" from their 1969 self-titled album:

Giorgio Moroder

the Godfather of Italian electro, Giorgio Moroder. A promotional TV video for the 1984 Olympics in LA:

R. Stevie Moore

live on NYC public access TV, the Scott & Gary Show in 1984.
"Showbiz is Dead" from Dumb Philosophy (1981):

"I've Begun to Fall in Love":

The Associates

the Associates, led by Billy Mackenzie, were one of the more successful new wave groups of their day; after touring with the Cure in 1980, John Murphy and Mike Dempsey (previously of the Cure) joined the group, and they landed album of the year in 1982 for their third album, Sulk. the following are sessions recorded for John Peel's radio show in April of 1981.
"It's Better This Way":

"Nude Spoons":

"Ulcragyceptimol":

William De Vaughn

diamond in the back, sun roof top, diggin' the scene with a gangster lean.
1974 single, "Be Thankful for What You've Got":

Sister Irene O'Connor

In the 70's, an Australian nun recorded an album using a drum machine and and a derelict synthesizer; her name was Sister Irene O'Connor and the album was Fire of God's Love/Songs to Ignite the Spirit (1976).
"Fire of God's Love":

"Mass-Emmanuel":

"Keshukoran":

Mikey Dread

Dread at the controls, 1979 with the album, African Anthem (The Mikey Dread Show Dubwise).
"Saturday Night Style":

"Industrial Anthem":

"Pre-dawn Dub":