Danny Elfman went on to be maybe the best film composer of the modern era.
Squeeze--my favorite of the era--the most critically acclaimed songwriting team of the 80's and were often compared to Lennon/McCartney for lyrical quality and musical variety.
Loved the Smithereens and Howard Jones...saw them both twice in New Orleans.
Not mentioned yet?
INXS (although we've had threads on them recently)