Smiths - "How Soon Is Now?", like darn near everyone else who gets to like them (it's as close to a hit as they ever had). I first noticed it in <i>The Wedding Singer</i>, I hate to say. >_<;; Then heard it on the radio during a "prehistoric" hour *stabs the new rock format with a pen*... and also a visitor to the site I had then, who I just thought was the coolest thing ever, was a really huge Smiths fan, and on his recommendation I picked up <i>Hatful of Hollow</i>. Yay.
U2 - "Hold Me, Thrill Me, Kiss Me, Kill Me". From the Batman ...Forever? something? soundtrack, which I bought with part of my graduation money (it was released the day I graduated from high school.

) I'd gotten it for "Kiss from a Rose", which had not yet been played to death, but HMTMKMKM impressed me as well. Plus the video reigns supreme, but I didn't know that at the time.
Depeche Mode - "Policy of Truth". Heard on the radio, picked up the <i>Violator</i> tape at the tender age of 15 having half remembered that song. Ah. Love. First step toward my current tastes, I think.
Erm... Erasure was "Run to the Sun", but no one's heard of them (only one minor dancey-pop hit that I don't even like [Alllllways, I wanna beee with you, and make belieeeeve with you, and live in harmony, harmony...] and "Chains of Love", a song from the 80s people only vaaaaaaguely remember) ...so. *cough* I'll skip 'em.
BNL was the entirety of <i>Rock Spectacle</i>, borrowed from my sister; Steve's voice did me in. *drools* And Ben Folds Five was "Battle of Who Could Care Less", also heard on the radio.
And that's most of the top ten, because I'm rambly.
