Sad songs are a bit of minefield with me because they can easily slip from being effective, emotionally, to being annoying and cloying. It often depends on how often I hear them. I love Enya's music to bits - she is one of my favorite living performers - but her Only Time was destroyed for me because of its extreme overuse in the wake of 9/11. I resented how it became the event's unofficial theme song and it colored Enya's career for several years after that. I'll never forgive Larry King for overlaying footage of the Afghanistan invasion over top of Enya performing "May it Be" from LOTR on his show -- a song with dick all to do with the Afghan war or 9/11.
And sometimes a circumstance will make a sad song actually dangerous. I'll will never listen to Holly Cole's cover of Tom Waits' "I Don't Wanna Grow Up" ever again -- nor will I ever listen to any of her other work -- because I heard the song playing on the radio the day I had some major career and personal upheavals occur in my life and it sparked the first - and thank god, to date the last - occasion in my life when I actually considered doing away with myself. It speaks to the power of music that it can actually be physically dangerous at times.
(BTW before anyone calls for an intervention, I should point out this happened more than 10 years ago and the circumstances that provoked the reaction no longer exist).
Interestingly, however, Waits' version has absolutely no effect on me whatsoever, probably because I find his singing voice to be so horrid that it overrides any feeling that might be generated by the song itself.
Now as for sad songs that don't make me want to put a gun in my mouth, one of my favorites is of all things an Elvis Presley song that hardly anyone has ever heard. Back in 1966 he recorded a version of Bob Dylan's "Tomorrow is a Long Time" , an epic 5-minute long ballad recorded at a time when he was usually recording 80-second-long movie jingles. It is an incredible performance, far outstripping 99% of his classics from the 50s. But RCA didn't know what to do with it so it was included as a "bonus song" on the soundtrack album of that all-time Oscar-winning (not) classic, Spinout. And it was never heard from again (well, it has shown up in some compilations, but I'd have much rather seen it get revived than A Little Less Conversation and Rubberneckin were a few years back. Try and find it if you can - if you're an Elvis-hater or think all he did was sing about Hound Dogs and eat peanut butter and banana sandwiches while shooting TV sets, it'll change your view of him. At least for that one song. According to the story, this was one of the only times Elvis actually told the Colonel and RCA to take a flying leap (which is probably why the song got buried - if it had been properly released his so-called Comeback might have been sparked a year or two earlier).
Cheers!
Alex