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":

Devo

Before the hats came the glasses and yellow suits. Devo was formed sometime around 1973 at Kent State as a punk rock/art rock combo. They DIYed a few singles on their Booji Boy label, then were signed to Warner Bros. in '76. Here are some videos from a live performance in France in 1978, the year they released their first album, Q: Are We Not Men? A: We Are Devo.
"Gut Feeling/Slap Your Mammy" from Are We Not Men:

"Smart Patrol/Mr. DNA" from their second album, Duty for the Future (1979):

a devolved version of "Red Eye" also from Duty for the Future:

Angelo & His Egos

obscure Greek new wave, Angelo & His Egos released one record, Egotrip (1982):
"Blackness":

The Short Wave Mystery

The only album ever released by the group was a four song EP of sorts called Pilots which was recorded while the members were in community college in California. 1000 copies were pressed, 500 of which were not even sold. the record, when for sale, can fetch up to $1000.
"Nice Girl":

"Ecaterina Szabo":

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":

Soft Cell

British synthpop duo best known for their hit single, "Tainted Love," Soft Cell with "Sex Dwarf" from Non-Stop Erotic Cabaret (1981). The song also had a banned video in the UK.

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):

Associates

"Skipping" from the 1982 album Sulk:

Vanity 6

"Make Up", the b-side from the 1982 single "He's So Dull":

Napalm Babies

"Don't Catch Me In the Blast" found on the White On Black (Experiments with Mass) cassette compilation (1982):

ed. 3/27/11: video embedding disabled, listen here: url: http://www.youtube.com/watch?v=1wJJzZt-2ZE

Pink Military

title track from Do Animals Believe In God? (1980):

Martin Dupont

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

Clan of Xymox

the goth genre was born on the cusp of the new wave movement, birthed out of some of its darker acts, and was often referred to as dark or coldwave. Clan of Xymox was one of the first bands to be given the title.
here is their second single, "A Day" (1985):

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"

Barnes & Barnes

"Cemetery Girls" from Voobaha (1980):

"Linoleum" also from Voobaha:

Pas de Deux

"Cardiocleptomanie" from Axe Ends (1983):

Cleaners from Venus

Cleaners from Venus was a DIY cassette band formed in England in 1980 helmed by singer/songwriter Martin Newell, who has continued to produce an enormous amount of recordings to the present day.
"A Blue Wave" from the first Cleaners from Venus cassette, Blow Your Troubles Away (1981):

"This Rainy Day" from Midnight Cleaners (1982):

"Ghosts in Doorways" from In the Golden Autumn (1983):