Is this one a breakup song? I alway thought it was a You’re So Damn Sexy That I’m Falling in Love With You Against My Better Judgement song.
Anyway, my contributions are
Boots of Spanish Leather by Dylan
and
Ain’t Life a Brook by Ferron
“Just when I think life’s a polished stone,
You give me a long drawn look,
I know pretty soon you’ll be leaving our home”
Depends on what you wish to express about the breakup, really.
Bonnie Raitt’s “I can’t Make You Love Me” is amazing, of course.
This is the song I actually sent to my ex via email when I realized we were truly over. It was my genuine feeling. Still is, despite everything, for which I’m eternally grateful.
Hootie and the Blowfish Goodbye is another sad one.
On the gutwrenching side of things, UK #1 “Dry Your Eyes” by The Streets (video) (lyrics). Nominally hip-hop, this is more like spoken word over acoustic guitars and strings.
What? No Somebody that I Used to Know? 
I like Black, by Pearl Jam.
Ooh! Ooh! How could I forget Different Drum???
Raul Malo, The Mavericks, I Don’t Care If You Love Me Anymore
Neil Sedaka Breakin’ Up Is Hard To Do
Eric Clapton Promises
Everly Brothers Bye Bye Love
Righteous Brothers You’ve lost that loving feeling
Cruel You by The Shoes.
Or pretty much any other song he’s written since '87.
A couple by Todd R: The Last Ride and It Wouldn’t Have Made Any Difference.
Another sad one, Someone Like You by Adele.
It was for me. And it has these lyrics: “What a wicked thing to say, you never felt this way” and “And I never dreamed that I’d lose somebody like you,”.
No mention of “Positively Fourth Street” by Bob Dylan?
Paul Simon, 50 Ways to Leave Your Lover
Pretty somber, really.
Clapton/Bonnamossa Further On Up The Road
The Clapton version helped me get through the big breakup in my life.
Heh,…I just started a thread for this one.
Here’s a favorite break up song of mine at the moment: Joshua Radin’s Winter.
Don’t Think Twice, by Bob Dylan
Everything About You by Ugly Kid Joe!