None of these are very imaginative examples - hooefully better ones come to me - but a few which spring to mind are In dreams by roy orbinson in blue velvet. That's the best one I think Where is my mind by the pixies in fight club Just like honey by jesus & mary chain in lost in translation I found all the use of music in reign over me strangely effective, and none of it was stuff I'd personally listen to
Head over heels by tears for fears and killing moon by Echo and the bunnymen in donnie darko. Not exactly a top notch movie or music but worked so well in context. Hey I'm a disaffected teenager in the 80s riding my bike at night