Granted, they’re an oft-maligned art form. But I like 'em, and come of age to 'em.
So, without any further ado, here are some of mine:
“Once in a Lifetime” - Talking Heads
“Nothing Compares 2 U” - Sinéad O’Connor
“True Faith” - New Order
“Believe” - Chemical Brothers
“Praise You” - Fatboy Slim
“Come to Daddy” - Aphex Twin
(There shall be more soon.)
Yours?
Tom Petty had some great ones back in the day
You Got Lucky - my personal favorite video of all time
Don’t Come Around Here No More
Into the Great Wide Open - featuring some guy who went on to become a famous actor, I’m told
Mary Jane’s Last Dance
Smooth Criminal - Michael Jackson. Can’t link while working, but if no one else does, I’ll link after work.
Gwen Stefani - ‘What You Waiting For?’
Michael Jackson - ‘Earth Song’
Madonna - ‘American Life’
Britney Spears - ‘Everytime’
Culture Club - ‘The War Song’
Eurythmics - ‘Beethoven (I Love To Listen To)’
Christina Aguilera featuring Redman - ‘Dirrty’
George Michael - ‘Freeek!’
Girls Aloud - ‘The Promise’
Lady Gaga featuring Beyonce - ‘Telephone’
Janet Jackson - ‘Rhythm Nation’
Kylie Minogue - ‘Did It Again’
Madonna - ‘Like A Prayer’
Madonna - ‘Bad Girl’
Madonna - ‘Bedtime Story’
Madonna - ‘What It Feels Like For A Girl’
Madonna - ‘Die Another Day’
Michael Jackson - ‘Smooth Criminal’
Michael Jackson & Janet Jackson - ‘Scream’
Michael Jackson - ‘Who Is It?’
Pet Shop Boys - ‘Can You Forgive Her?’
Spice Girls - ‘Viva Forever’
I love music videos. This list skews more towards the pop side of my music listening but I think pop acts generally make better music videos than artists in the other genres I like.
A-Ha - Take On Me
Depeche Mode - Enjoy The Silence (the animated video for the remix of the song)
I also can’t like here, but they’re easy enough to find.
Peter Gabriel’s Sledgehammer. There’s something about dancing naked headless chickens that’s bizarre beyond belief.
My absolute favorite is Nine Inch Nails’ The Perfect Drug. I’ve always wished to see a movie just like that, but so far that sort of stylized gothicism hasn’t shown up in anything full-length that I’ve ever seen.
Others I like:
Stabbing Westward “Shame”
No Doubt “Don’t Speak”
Vitamin C “The Itch”
Ash “Girl From Mars”
She Wants Revenge “Tear You Apart”
Bat For Lashes “What’s A Girl To Do?”
Marilyn Manson “Tainted Love” (Grey’s Anatomy fans, watch this one)
Jamiroquai - Virtual Insanity, as ever.
For all you red-blooded men out there:
[spoilered NSFW link]
A pretty samey version of the William DeVaughan original, but with an interesting video. There’s a long preamble, the show starts about 1min 30secs.
I once put the Massive Attack Collected DVD on and walked out of the room absent mindedly… only for my mother to walk in just as this video was playing.
Awkward.
Freewheel Burning - Judas Priest
Most of my favorite videos are from the '80s. I’m sure I’ll come up with more after I post this, but these are the ones I can think of right now.
“Sweet Dreams (Are Made of This),” Eurythmics - I have a mad girl-crush on Annie Lennox in this video, and have since I first saw it in the 80s.
“Right Now,” Van Halen
“Don’t Pay the Ferryman,” Chris DeBurgh (can’t find this one, didn’t look too hard)
“Sledgehammer,” Peter Gabriel
“Hungry Like the Wolf,” Duran Duran
“Take On Me,” a-ha
“Jeremy,” Pearl Jam
“Billie Jean,” Michael Jackson
“Safety Dance,” Men Without Hats (I also like the literal videoversion) 
Muse’s Knights of Cydonia. Hands down.
A bunch of my favourites have been mentioned already…some which haven’t…
David Lee Roth - Just a Gigilo/I Ain’t Got Nobody
George Harrison - Got My Mind Set On You (The version that doesn’t take place in his haunted study.) Unusual in that I only discovered it recently, when someone linked it in a thread here… Alexis and the girl have a great interplay, and George’s bits are delightfully oddball.
Quite a few from Tomoko Kawase’s various projects…Some specific examples…
Tommy heavenly6 - Lollipop Candy 
BAD:heart: Girl (Get a snack, the song is 10 and a half minutes long, but the short version of the video misses most of my favourite bits.)
Tommy heavenly6 - Heavy Starry Chain
Tommy heavenly6 - I’m Gonna SCREAM+
Tommy heavenly6 - Wait 'Till I Can Dream
Tommy february6 - Bloomin’
Tommy february - Magic in youR Eyes
the brilliant green - CALL MY NAME
the brilliant green - Enemy
cough OK, enough of my Tommy-pimping…
You Can Call Me Al - Paul Simon/Chevy Chase
Addicted To Love - Robert Palmer (ah, memories)
Forgot Simply Irresistable. I like the dancing models.
Primus - Winona’s Big Brown Beaver
Ah, the 80’s, best years of my life! I actually do have a bunch of MTV videos on videocassettes somewhere, I wonder if they’re still playable?
I liked everything by David Bowie, Ashes to Ashes especially.
Everything by Adam Ant!
Everything by Devo!
And early in the MTV video days, I remember a song by Kate Bush, The Man With the Child In His Eyes, but it didn’t seem to ‘fit in’ and I thought, well, that’s awkward. I only saw it maybe three times. But that damn song by Kate Bush grew on me, I couldn’t forget it, and that, friends, is how I got to appreciate Kate Bush. (her talent was so obvious).
If your mother knew the sheila used to be a bruce it might have been even more awkward.
Funny video, but is Roth lip-synching the Louis Prima song or actually singing it? It sounds too exactly the same to be the latter.