“The Stroke” by Billy Squier
I see your goth anthem and raise you “The Cure - The Hanging Garden.” 
I was thinking Wipeout too…
Foo Fighters - My Hero
all right sorry i’ve blanked
California Uber Alles-- Dead Kennedys
Aja, Steely Dan. One of the finer performances by Steve Gadd; done in two takes.
Close To You, The Carpenters. Hal Blaine didn’t touch the snare drum once in this song (unless you count the album version; a snare drum, albeit muffled, can be heard in the encore).
Wipeout, The Surfaris.
Come Together, The Beatles.
Once upon a time, Smells Like Teen Spirit by Nirvana also held this distinction.
My Sharona by The Knack
Chris W
Metallica’s “One” is all about the machine-gun double bass rolls. That’s the key to that song.
In similar fashion, drums are the key to “Creeping Death” and “Frantic.”
Meat Loaf’s * Life Is a Lemon (And I Want My Money Back.*
Fire by Jimi Hendrix has a great drumbeat courtesy of Mitch Mitchell
Closer by NIN.
Janies Crying by Van Halen
Badtz Maru wrote
Hey! I already claimed Wild Thing above.
YYZ by Rush
Not super well known, but Leave My Monkey Alone by Warren Zevon (produced by George Clinton, which explains alot about it’s sound)
oh, and I’m surprised I’m the first to mention
Blue Monday by New Order
Hey man, I already covered that 
Rhymin and Stealin - Beastie Boys
Hard to Explain - The Strokes
In the OP, I had meant to disqualify songs like “Moby Dick” and Rush’s “The Rhythm Method”, which are essentially extended drum solos.
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Great calls on “Addicted to Love”, “50 Ways to Leave Your Lover”, “Tusk”, “Janie’s Cryin’”, “One”, “In the Air Tonight”, and “I Don’t Care Anymore”, and “We We Will Rock You”.
How about James Brown’s “Funky Drummer”? This is probably one of the most sampled pieces of music around.
As I read the thread title - ‘Making Plans for Nigel’ XTC - I got there first?
“Need You Tonight” by INXS
The theme song from Hogan’s Heroes
Haven’t heard this one in a while … doesn’t “Rhymin’ and Stealin’” sample the drumbeat of “When the Levee Breaks”?
Hey Mickey