As a music collector & college DJ, my partner & I would have different themes. We did a show once on songs that didn’t say (sing) the title in the lyrics. Of course instrumentals don’t count. Examples are,"For What it’s Worth " by Buffalo Springfield, & “Unchained Melody” by Righteous Brothers. I recall we also eliminated Dylan-he had too many.
Any others?
VALOTTE by Juian Lennon
MERCENARIES by Harry Chapin
Day Of The Lords by Joy Division
How Soon Is Now by The Smiths
Uno by Muse
Talk Show Host by Radiohead
Song 2 by Blur
“Chop Suey” by System of a Down
“Hazard” by Richard Marx (might have to double check that one)
Many Nivana Songs (eg “Smells like Teen Spirit”)
The Pearl Jam song about the old woman in a small town (Forgot the title, but it’s not in the song).
Krokodil, it’s called Elderly Woman Behind The Counter In A Small Town by Pearl Jam
Black Dog by Led Zeppelin
Brain Damage by Pink Floyd
An Cat Dubh by U2
Damn you, Necromancer! “How Soon Is Now?” is the song that immediately popped into my head upon viewing the thread title. 
I’ll offer “Flake” by Jack Johnson instead.
From the CD’s I have in the room with me:
A Day in the Life-The Beatles
Dreamline-Rush
Mothers of the Disappeared-U2
ReJoyce-Jefferson Airplane
Eat Starch Mom-Jefferson Airplane
Traid-Jefferson Airplane
Paintbox-Pink Floyd
Itchycoo Park by Small Faces
The Ubiquitous Mr. Lovegrove, by Dead Can Dance
Standing Outside a Broken Phonebooth With Money In My Hand, by Primitive Radio Gods
Bizarre Love Triangle, by New Order
Ode to Boy, by Yaz
“Sukiyaki” by Kyu Sakamato
“Opus 17” by the 4 Seasons (the song that goes "I’ll be strong, I’ll try to carry on, etc).
“Lover’s Concerto” by the Toys (the song that everyone thinks is called “How Gentle Is The Rain”)
Incidentally, the Small Faces do mention the title of the song in the second line “under dreamy skies, to ITCHYKOO PARK that’s where I’ll be”
8STOPS7–Question Everything
Eclipse—Pink Floyd
The Ballad of John and Yoko—The Beatles
Iris—The Goo Goo Dolls
Duvet–Boa
I’m sure there’s more, maybe even on that cd, but I’m tired and don’t want to open up the playlist to listen.
Actually Opus 17 has as part of the title(Don’t You Worry Bout Me), so that wouldn’t count. You’re correct about Itchy Koo Park,wolf. I didn’t mention our show focused on oldies-50’s to mid- 60’s; later it obviously is far from rare, much more common than I thought.
But I am very impressed by the Dopers.
Duvet is an excellent song, I bought that album after hearing it used as the theme song for Lain… which is my way of sneaking in Cool: World Without Logos, the theme song for Hellsing. Actually I have next to no idea what the lyrics say, even though they are in English, but I’m pretty sure they did not say Cool. (sorry, this may not count as I’m pretty sure it didn’t get radio airplay)
I keep trying to think of an Allman Brothers song that fits. There aren’t many (among those with words, anyway). Revival does, though.
IIRC, the last line of the first verse is: “To Itchycoo Park, that’s where i’ve been.”
Darwin’s Finch: The Ubiquitous Mr. Lovegrove was one of the first songs that came to mind when i saw this thread. Gotta love Dead Can Dance.
Blue Monday - New Order
Love Song - The Cure
Knowledge - Operation Ivy
Brain Stew & Longview - Green Day
White Man in Hammersmith Palais & Train in Vain - The Clash
Man, am I embarrassed. Thanks for pointing out my error concerning Itchycoo Park. I’ll have to listen to it more carefully.