Is That All There Is?: Peggy Lee. Not merely sad, but downright depressing. It was popular around 1970, and it put me in a really dark mood whenever it was played.
I second Danny Boy.
A lot of classical music moves me to tears, but not necessarily out of sadness. The third movement to Tchaikowsky’s Sixth Symphony is positively gut-wrenching.
Kindertotenlieder (Songs of dead children) by Gustav Mahler, especially when Dietrich Fischer-Dieskau sings them.
All the Way from America[sub]1[/sub] -Joan Armatrading
The Partisan and Who by Fire[sub]2[/sub] -Leonard Cohen
Way to Blue[sub]3[/sub] -Nick Drake
Small Town Boy[sub]4[/sub] -The Bronski Beat
Anyone who has ever had a long distance relationship, or has one now, will cry at this song.
The lyrics of the first song, “The Partisan” are partially in French. The ‘mood’ of these songs can make me cry.
Makes me think of the sadness in a young musical genius’ heart that drove him to take his own life.
The theme song for the ‘Angry Young Man’ circa 1985, when I knew a particular Angry Young Man who also considered it.
Sorry, make that 1984.
Glad Nick Drake made an appearance.
Oh man, this is easy. The Saddest Song in the World[sup]TM[/sup] is The Ballad of John Henry by Johnny Cash. That song does something to me that I can’t even describe.
Tom Trouberts Blues (4 sheets to the wind in Copenhagen) by Tom Waits.
When Queen recorded Innuendo (1990/1991), Mercury was in the last stages of his decline. I don’t think his illness was known in 1986, when A Kind of Magic was released.
Bobby Goldsboro, and… ye GADS. What a horrible song. Not sad, just… ick. 
My own suggestions.
The Pogues version of And the band played Waltzing Matilda has to top the list.
A good runner up is Pink Floyd with The Gunners Dream.
Not mentioned before in this thread: 41 Shots (American Skin) by Bruce Springsteen. Of course, there’s a ton of songs by the latter that fit the bill, like The River and One Step Up. But 41 Shots, especially off the “Live in New York City” CD? I dare you to listen to it without choking up.
All these posts and no one’s mentioned:
Alone Again, Naturally by Gilbert O’ Sullivan
or
Long Black Veil (many artists recorded this, don’t know who first)
?!?
Have Yourself a Merry Little Christmas is terribly melancholy, I think. And Beth Neilson Chapman’s Sand and Water is very sad.
StG
Eric Clapton’s Tears in Heaven.
Taps. Not for any specific reason, but it sounds so mournful.
China - Tori Amos
A Little Fall of Rain - from Les Miserables (oh god… it gets me just thinking about it)
AND Operator by Eddie From Ohio, a folk band from the Washington, D.C. area, is THE saddest song EVER.
That new song by Emimum about his mother… makes me sad. It is an angry song… but it makes me sad about what passes for popular music.
Requiem: Gyorgy Ligety
Requiem: Wolfgang Amadeus Mozart, especially Lacrimosa
I think this thread needs a Reader Advisory: Contains Deeply Melancholy Lyrics.
I only know a few of these songs, but reading through the lists is like: "Ohh I know that one… " 
sniffle… “And that one too…” sniffle… sniffle… And then I made the mistake of looking up the Lyrics for And the band played Waltzing Matilda… dear god, somone just pass me a razor.
My own personal picks have been covered… and I was very pleased to see someone else loves (?) When the Tigers Broke Free.
I didn’t see Amazing Grace on the list so far – not technicaly a sad song… but I guess when you play it at every funeral in the family it’s likely to acquire a little baggage.
Ohh… and An Arky’s comment reminded me: The Foggy Dew. (I have the Chieftains The Long Black Veil disc which has both tunes on it).
The songs that always get me, for various reasons, are:
High Hopes by Pink Floyd. I can’t listen to this song when anybody’s around because it always makes me cry.
Oh Life (There Must Be More) by Alan Parsons. Same thing. It’s based on a true story about a woman who committed suicide.
Brother Up In Heaven by Alan Parsons
Run for the Roses by Dan Fogelberg. I don’t even know why this one makes me cry, but it never fails.
Pussy Willow by Jethro Tull.
Roses for Mama by C. W. McCall. Glad to hear I’m not the only one.
Padriac My Prince by Bright Eyes. The song of a distraught mother who is contemplating suicide on the anniversary of her little child’s death…Sung tragically from the POV of the deceased infant’s brother.
Bill Evans, Some Other Time
Very melancholy. You wouldn’t think one man and a piano could be that expressive. No lyrics required.
Speaking of which…
Some Other Time from ON THE TOWN, the Bernstein and Comden and Green musical. Inexplicably cut from the film version along with almost all the other decent songs.
"Twenty-four hours can go so fast,
You look around, the day has passed.
When you’re in love,
Time is precious stuff;
Even a lifetime isn’t enough.
Where has the time all gone to?
Haven’t done half the things we want to.
Oh, well, we’ll catch up
Some other time…"
Avoid the film soundtrack and get the OCR with Nancy Walker et al, or the 1992 reconstruction with Frederica Von Stade, Thomas Hampson, David Garrison, and Tyne Daly.
All the more melancholy in the original, which opened the day after Christmas, 1944. These boys may indeed have been spending their last carefree day of their lives, as they were sailing the next morning to Lord knows where.
Add Softly, As I Leave You, sung by Frank Sinatra. That one really brings me down.
Me And Bobby McGee, sung by Janis Joplin can make me at least nostalgic, if not sad.