Primal Scream

If there's a style of music Primal Scream hasn't tried... just be patient, they'll probably get around to it.

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.

Shriekback

Alt Goth Funk?  Shriekback are cool, sublime, sometimes dancey, sometimes dark.

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.

Magazine

Magazine are the archetypal post-punk band. Started by Howard Devoto after he left the Buzzcocks, they combine the energy of punk with slower tempos and interesting guitar work. Sometimes punchy, other times dark and brooding, their first 2 albums are essential.

Left of the Dial: Dispatches from the '80s Underground

Probably my favorite '80s box set. Are there big artists left off? Yes. Are there some songs I would've swapped out for different ones? Yes.  But as an overview of what alternative music was in that decade, this pretty much touches all the different sub-genres.  If there is one single item you could give to someone and say - this is what alternative music was in its first decade - look no further.