“Man in the Long Black Coat”- Bob Dylan
“Who Are The Brain Police”- Mothers of Invention
For some reason, the first thing that came to mind was Red Right Hand by Nick Cave and the Bad Seeds.
“With Her Head Tucked Underneath Her Arm,” by The Kingston Trio.
It’s about Anne Boleyn. 
Nuages (clouds), nocturne no. 1 for orchestra by Claude Debussy.
Agnus Dei/Lux Aeterna/Requiem Aeternam from Gabriel Faure’s Requiem.
I’m a little surprised no one has mentioned D.O.A by Bloodrock yet.
Different songs haunt me in different ways.
A few that come to mind:
Handel - Sarabande
Arvo Pärt - Spiegel im Spiegel
Peter, Paul and Mary A Soalin’
Zupfgeigenhansel Andre, die das Land so sehr nicht liebten
Frühling dringt in den Norden
Gustavo Santaolalla - Secreto en la montaña
Vienna Teng - The Hymn Of Acxiom
Ich Hatt Einen Kameraden
Beethoven - Allegretto from Symphony No. 7
La bruja- Tlen Huicani
Gustavo Santaolalla - Pa’ Bailar
I struggle with threads like this that deteriorate to lists, even if they have YouTube links.
Where is the “why” behind the choices? I would much rather hear one recommendation - your best, something interesting, whatever - with some of your thoughts on it.
I will take a shot:
This is Music for 18 Musicians by Steve Reich. It is Minimalist - played on orchestra instruments, but they all play little note-patterns, which are written to weave in and out with each other over time, like listening to rain drops off different parts of your gutters and the plops run in and out of sync with each other. Philip Glass is probably the best-known minimalist composer. It was revolutionary in the '60’s and influenced music, including Pete Townshend’s synth work on Who’s Next (he typically name-checks Karlheinz Stockhausen)
Why haunting? It is not music as much as a sonic landscape. It starts off sounding totally mechanical, but as you hear the note patterns weave in and out, it starts to sound like…breathing? And the Reich adds in vocal elements to increase the non-mechanical feel.
The sonic landscape feel of the piece is immersive, when you get lost in the patterns you go places you don’t go with other types of music. Sorry, that is the best description I’ve got ;).
What is haunting about this is that, once you get this feel, it stays with you. If your brain flashes on something that gets Music for 18 Musicians going in your brain, you can be transported back to that place. The feel of the patterns stay with you. Freaks me out sometimes 
So that’s my Why. Worth a try!
Haunting as hell… and I don’t even know what they’re saying:
Pink Floyd’s “Wish You Were Here.”
This is one of my favorites. Really unusual for 1963.
I think of it as a folk song, although it isn’t. There are, of course, truckloads of folksongs that are haunting, both in the subject matter and in the musical setting… some really traditional, like “The Silkie of Sule Skerry” and “Cruel Sister”, which deal with death and the supernatural, and others more recent, like “Morning Dew”, which is about nuclear apocalypse.
Another bunch of haunting songs came out of the West Coast sound of the late 60s, in which a mysterioso feel was one of the major elements. (It’s no coincidence that a lot of the West Coast musicians started as folk artists.) The Doors have already been mentioned (the song that first came to mind for me was “End of the Night”)… Jefferson Airplane had some very haunting songs, both sweet (“Comin’ Back to Me”, “Today”) and scary (“The House at Pooneil Corner”). Another favorite of mine is the trancelike “Bulgaria” by It’s a Beautiful Day.
Oh, and another: Donovan Season of the Witch. There are other good versions, but he wrote it.
The Sound of Silence by Simon and Garfunckle.
I’d choose Pink Floyd’s, The Wall as the most haunting album—all of the songs are equally disturbing, yet beautiful.
You can’t get much darker of a theme than tracking a man’s degeneration into insanity. The tone of the total package is surreal, but believably real at the same time. The music, lyrics, inter-cut dialogue/sound bites, even the movie animation, all come together to make this a chilling, unforgettable masterpiece.
…not a good album to listen to if you’re depressed, however.
While we’re on the subject of Pink Floyd, I find “High Hopes” (from The Division Bell) to be quite haunting as well.
Good one.
**The Division Bell **seems to get panned in some quarters, but I’ve always liked it quite well.
And Mother is SOOOOO creepy. 
I just don’t respond as emotionally to it as I do to Wish You Were Here. Of course, you have to know its history to really get it.
In Dulce Decorum by The Damned
Orbison’s “In Dreams” is also ruined.
Ironically, a love song written in 1937 has always haunted me: Lili Marleen.
Vera Lynn’s version; Marlene Dietrich’s version (this is the one that affects me most).
The song was popular with both Allied and Axis soldiers in WWII’s European Theater. A love song about a soldier and the girl he yearns for, listened to by soldiers on both sides during a devastating war, knowing many of those listeners would be going back to their girls in body bags is sobering and haunting.
Although I wan’t born yet, WWII affects me on an emotional level. My father was in the Army Air Corps (385th Fighter Squadron of the 364th Fighter Group) and recounted how many of his friends did not return from their P-38 and P-51 missions, or crashed and burned on their return to base. My mother witnessed even more horror, seeing her neighbor blown into the next block by a doodlebug and being being bombed out of three houses during the war, the last one killing her dog and nearly killing her).