If you can find the video on YouTube or a similar site (GoogleVideo?), that’s good, but a linked video is not required.
My favorite videos are (a lot of these are NSFW):
UNKLE - Eye For an Eye
Aphex Twin - Come to Daddy
The Prodigy - Smack My Bitch Up
Fatboy Slim - Ya Mama (They call it “Push the Tempo” on YouTube but that’s not the title)
Bjork - Bachelorette
Those are the only ones I can think of now so there may be more…
Oingo Boingo- Nothing Bad Ever Happens to Me.
“Weird Al” Yankovic - Fat
Not only for being a dead-on clone/parody of Michael Jackson’s “Bad” music video, but also because Al filmed it on the same subway set Jackson used. Booyah! 
Fatboy Slim - Weapon of Choice
That’s one of my favourites too.
I also like All in my head by Kosheen, but I’m not sure why.
Aphex Twin - Windowlicker
Man, there were a lot of great ones when MTV used to show videos. The crappy video quality on Youtube doesn’t do them justice.
Beastie Boys - Sabotage http://youtube.com/watch?v=lzTxuDUfzzM
Nine Inch Nails - Closer (NSFW!) http://youtube.com/watch?v=e0oLe31r7B4
Tool - Sober http://youtube.com/watch?v=c7knw8lsIvI
Metallica - Unforgiven (this one really creeps me out) http://youtube.com/watch?v=o5tnQrIiVXs
REM - End of the World as we Know It http://youtube.com/watch?v=4UT14WZk_ho
Peter Gabriel - Sledgehammer http://youtube.com/watch?v=NaPYQysTHtA
Guns N’ Roses - Sweet Child o’ Mine http://youtube.com/watch?v=oZgMFHbmYiE
Nirvana - Smells Like Teen Spirit http://youtube.com/watch?v=_1Qdcdw22Sw
Van Halen - Hot for Teacher http://youtube.com/watch?v=vJVwqKhaEbQ
Beastie Boys - Fight for Your Right http://youtube.com/watch?v=M5LO2VWJaV8
Jane’s Addiction - Been Caught Stealing http://youtube.com/watch?v=8KMrIZRg9B8
Ok I’ll stop now. Man, Youtube is cool.
Metallica “One”
Cyndi Lauper “Money changes everything”
Rust HDistant Early Warning"
I think we did this not so long ago…
Anyway,
Madonna - Like A Prayer
Madonna - Frozen
Röyksopp - What Else Is There
Sophie Ellis-Bextor - Murder On The Dance Floor
Sophie Ellis-Bextor - Music Gets The Best Of Me
Björk - It’s Oh So Quiet
Björk - All Is Full Of Love
Shakespeare’s Sister - Stay
Utada Hikaru - Keep Tryin’
Sinead O’Connor - Nothing Compares To U
Koda Kumi - Cutie Honey
Blondie - Rapture
Eurythmics - Love Is A Stranger
Buggles - Video Killed the Radio Star
Michael Jackson - Thriller
And now I’ve reached a mental block, as I can’t seem to think of any more. Oh dear.
Pearl Jam’s Do The Evolution
An absolutely spectacular video. It should scare you.
Two by Tom Petty and the Heartbreakers
Don’t Come Around Here No More - really much more creepy than I remembered
You Got Lucky - best video ever made
Back from an era wen a “music video” was more than just a film of the band playing on stage. In fact, other than 15 seconds of a guitar solo, we don’t see anyone in the band touching an instrument in either film.
I don’t mind doing this again, as I wasn’t signed up yet last time.
I’ve always liked the creepy Spitting Image puppets in Land of Confusion by Genesis.
Another favorite is the (previously mentioned) Sledgehammer video, because of the work Aardman (Wallace & Gromit) Animation did in it.
I also have very fond memories of Susanna Hoffs’ eyes in Walk Like an Egyptian by the Bangles, though I don’t much care about the rest of the video.
Ruh-oh. Looks like Clockwork’s double posting.
Pat Benatar - Love Is A Battlefield
The Pinker Tones - Karma Hunters
Cyndi Lauper - The Goonies ‘R’ Good Enough
Culture Club - Karma Chameleon
Actually… that last one really isn’t as good as I remember it being. :-\