Nitzer Ebb

This underappreciated band definitely walked the line between industrial and new wave. "Body music" indeed.

Big Country

Big Country never got the recognition they deserved in the US but still remained a big band in the UK and Europe until the end of their career in the late '90s.   Their first 2 albums were produced by Steve Lillywhite, who also worked with U2 and Simple Minds.  Thus, they got lumped in with those two bands, as well as The Alarm, as a new wave of anthemic rock of Gaelic origins.  They were actually the most seasoned and musically proficient band of that group, right out of the gate as heard on their debut, "The Crossing."  They went on to make great albums (except No Place Like Home) for the rest of their career until lead singer Stuart Adamson's tragic suicide in 2000.

Gary Numan

Gary Numan combined his early punk roots with some synthesizer and took on a "Man Who Fell To Earth" persona.  Sometimes it sounds like he made the same album 3 or 4 times in a row before evolving into something else.  But at 60 years old, he's managed to not become a parody of his early image, nor stagnant musically, which is quite a feat.

The Waterboys

Long-running Irish rock band led by Scotsman Mike Scott.

Alphaville

Best known for their 2 hits "Forever Young" and "Big in Japan," listeners may be pleasantly surprised by digging deeper into their catalogue.

Glass Tiger

Whether you find Glass Tiger to be representative of the worst middle-of-the-road 80s music, or hail them as one of Canada's best 80s bands... I dunno, but there is some enjoyable stuff among their 3 albums.

Ultravox

Ultravox never really found consistent success in the US, but did quite well in the UK and Europe.  They have elements that fit into post-punk, new wave, and prog rock, which you might think would mean they would have garnered a sizeable crowd, but those fickle Americans just wanted them to tick 1 box.  Their early years were fronted by John Foxx, then by Midge Ure in the '80s, both of whom went on to their own modestly successful solo careers.  Definitely an underrated band considering their longevity.