The Zombies were great.

I Believe in You - Talk Talk

Story of the Blues (Part 1) - Wah!

Another Tear Falls - The Walker Brothers

Just Say Goodbye - The Walker Brothers

The Sun Ain't Gonna Shine Anymore - The Walker Brothers

damien rice does an amazing cover of creep best iv ever heard!!!!

a few personal favorite sad depressing songs that im not sure anyone has already mentioned and not all classed as "indie" but still give me that lump in my throat!

Bright eyes - its cool we can still be friends
damien rice - 9 crimes
freightened rabbit - poke
the moody blues - nights in white satin (for when your feeling sorry for yourself)
Elbow - Friend of ours (about his best friend who died)
glen hansard and marketa irglova - the hill
Glen Hansard - Everytime (britney cover....trust me on this one!)
Kelly jones - nothing compares to you (sinead o'connor cover..beautiful)!
Eliott smith - between the bars
Swell season (glen hansard) - in these arms
Beck - already dead
Radiohead - exit music for a film
Amanda palmer - i will follow you into the dark (deathcab for cutie cover)
mumford and sons - liar
Jerry fish and the mudbug club - where the sun dont shine (personal favorite)
Villagers - on a sunlit stage
Bright eyes - for you
imogen heap - hide and seek
snow patrol - you could be happy
coldplay - the scientist
Jams taylor - fire and rain
Johnny cash - Hurt
Gary jules - mad world
coldplay - fix you
goo goo dolls - iris
the beatles - yesterday
john lenon - imagine
Damien rice - the blowers daughter
nick cave and the bad seeds - love letter
the verve - bittersweet symphany

iv hoads more but il be here all night in dat case!
all thes songs make me wana cry!

Sister Morphine - Marianne Faithfull
Where Was Your Heart (When You Needed It Most) - Soft Cell
I Wish I Never Saw the Sun Shine - The Ronettes
Two Worlds Collide - Inspiral Carpets
Rhymes of Goodbye - Scott Walker
Life Becoming a Landslide - Manic Street Preachers

Listen to BODYBAG by THE FILMS. An awesome band overall.

Oh and I have to agree here - gives me goose bumps

no mention of carissa's wierd is a shame. i'm a fanboy (they're my avatar and sig), but some of their most heartbreaking songs make death cab for cutie look like ABBA

so you wanna be a superhero
september come take this heart away
you should be at home here
brooke daniels' tiny broken fingers
low budget slow motion soundtrack song for the leaving scene
all apologies all smiles, yours truly, ugly valentine
you should be hated here

and many more

other breakers of my heart

casiotone for the painfully alone - killers
elliott smith - pitseleh
the national lights - swimming in the swamp
the smiths - i know it's over
galaxie 500 - tugboat
marissa nadler - diamond heart
smog - red apples
sun kil moon - carry me ohio
red house painters - katy song
the mountain goats - pale green things
deer tick - smith hill
sunset rubdown - the empty threats of little lord
jesca hoop - angel mom
parenthetical girls - stolen children
neutral milk hotel - oh comely/two-headed boy pt. 2
wye oak - if children were wishes
nick drake - place to be
iron & wine - sodom, south georgia
polly scattergood - i hate the way

Gwesy, you mentioned Love Letter by Nick Cave and the Bad Seeds and I completely and totally agree with that cdhoice... beautifully penned song, very mournful, regretful lyrics, just the epitome of a sad song, by the Master himself!

I adore Indie music, and can def reccomend you "Paris" by Friendly Fires....it's not really sad, but it's really romantic and I even cried a few times)

so, listen to it if you haven't yet)

as for the saddest song, listen to Emiliana Torrini "Serenade" or Kings of Convenience "Peactime"