The Smiths – "I Want the One I Can't Have"
Sure, any Smiths song will do, but this track, from their 1985 album Meat is Murder, expresses the ache of longing better than any. As a teenager I always interpreted it as being kind of, well, gay, but now I realize it's pretty universal:

Marvin Gaye – "I Heard It Through the Grapevine"
While a lot of Motown songs deal with heartbreak, most of them are so irrepressibly bouncy (see "Where Did Our Love Go") that they don't seem to really feel it. Marvin Gaye feels it, though, and this may be the penultimate "holy crap I just found out you're cheating" song:

The Streets – "It's Too Late"
While British rapper Mike Skinner was well-known for his jaunty beats and intricate witticisms, he also mastered wistful melancholy on tracks like "Blinded by the Lights" and this, from 2002's Original Pirate Material, that presaged the current dark dubstep sound a la Burial:

Beck – "Nobody's Fault But My Own"
As Jezebel points out, basically all of Sea Change is about heartbreak, but this track from 1998's Mutations always gave me a lump in my throat, and Beck's live version (accompanied only by a harmonium) is even more haunting:

Sinead O'Connor – "The Last Day of Our Acquaintance"
Of course, she's known for "Nothing Compares to You," one of the most popular breakup songs in history, but while that song's intensity owed a bit to Prince, O'Connor can also write a real tearjerker herself, and this track's understated matter-of-factness makes it all the more devastating.

Snif. Is 3:00 too early for a nice vodka drink?