First thing it came to me was Radiohead - How to dissapear completely and The Smiths Never had no one ever for instant depression! :)

Maybe not exactly indie in the sense you're asking for though.

sad song
 

i can feel a hot one - manchester orchestra
the light behind your eyes -mcr
the world is ugly - mcr
therapy- all time low

Be Quiet and Drive -Deftones

That song just kills my soul, if you haven't had the chance you should listen to it. It is extremely sad.

Off the top of my head I have a few random references for a variety of sad listeners. Whether you're dumped, cheated on, lost, doing hard drugs, isolated, or just seasonally moody I believe I have you covered. I'll come back if I think of anything else.
(Radiohead) Radiohead - There, There (The Boney King of Nothing)
(Early 00s Mildly Poppy Indie) Hot Rod Circuit - At Nature's Mercy
(Late 90s Sad Math Rock) Braid - Forever Got Shorter
(Folk Emo) Owen - A Bird In Hand (OR American Football - The Summer Ends)
(Relevant Shoegaze) Whirr - Formulas and Frequencies
(Surprisingly Miserable Melodic Club Music) RAC - If You Forget Me
(70s Sad Holiday) Joni Mitchell - River
(00s Mainstream Punk) Sum 41 Pieces
(Contemporary Lo-Fi Dream Pop/16 Candles on Prozac) Procession - Spring Thaw
(Late 90s drop-tuned misery) Deftones - Digital Bath
(00s indie/emo boiz) The Early November - Everything's too cold, but you're so hot

Ryan Adams - La Cienega Just Smiled... Or basically anything by Ryan Adams haha.

The Veils - Vicious Traditions

Band of Horses - The Funeral

I need some help, please.
I'm looking for a sad pop britpop alike song that I used to hear on radio between 00's to 05's. I remember that it had a piano intro very similar to Snow Patrol chasing cars or "the lightning strike" but a little bit slower and depressive but is not Coldplay. I also tried The Fray.
About Lyrics... sorry but it was 10 years ago and I didn't know that much how to "listen" lyrics in English. (Sorry about my English)
I recorded something that could help about this "piano intro" (youtube).
DdE4mKZAq78

If you want to hear sad songs, listen to Keaton Henson's 'Dear...' and 'Birthdays' albums. Brilliant, but so purely heartbreakingly emotional and oftentimes dark and sad.

Red light by the Rumour Said fire

The Smiths - This Night Has Opened My Eyes

no surprises by radiohead always gets me in the whole swandive out the window mood xxx

Kettering by The Antlers is one that I like to listen to it when I'm in that sort of mood.

There's a video on youtube of Little Motel by Modest Mouse, I don't think it's the official video but it's literally the saddest thing ever.

Kristin Hersh
Me and My Charms
You can't leave me now
I haven't left you yet

Red House Painters
Drop
i'd like to come home to see you
and embrace your illness under soft light
but then you'd know how much i really need you

Aimee Mann
It's Not
so baby kiss me like a drug, like a respirator
and let me fall into the dream of the astronaut

Cat Power
Colors and the Kids
must be the colors and it must just be the kids
that keep me alive on this January night

I'd say Steven Wilson-Routine so depressed :(

^^^ Several Steven Wilson songs are serious contenders for this honour, but I think it would be hard to go past:

Gilbert O'Sullivan - Alone Again... Naturally

Another one I could think of is "I'm not gonna miss you" by Glen Campbell, the context is just depressing enough to the make the song a hallowing cry for help.

Jaymes Young - Moondust

Sweet song classic

I almost cry, or actually cry, when I hear Snow Patrol - Run, because of the associations my mind makes to the text, and to the strength and power of the song.

Maybe it's just me, though.

The Smiths - Asleep

Phantasmagoria in two (Tim Buckley, 1967) and Katy Song (Red House Painters, 1993).

Skeleton Tree - Nick Cave's new album.
 

Nick Cave's new album is very emotional, as it's mostly about the death of his 15 year old son last year.
Great album - definitely the best album of 2016.

After I heard 'Jesus Alone' I'm to scared to listen to this album. It promises be too heart-braking.

Fleetwood Mac - Storms
Daughter - Switzerland
Bon Iver - Moon Water

What about Mazzy Star Fade Into You?

Like if you cry evry time. :(

Not very indie, but Heather Nova - Gloomy Sunday is the one for me. By the way, there many more versions of this song, including the one by Diamanda Galas, if Heather Nova is too pop for you.

The Sweetest Kill- Broken Social Scene
Asleep- The Smiths
Jamie Marie- Girls
Studio 6- Matt Maltese

Current 93's recitation of Thomas Ligotti's poem, I Have a Special Plan for This World is the saddest song on the planet. You will probably kill yourself after listening to it.

It begins,

and my favorite verse states:



/thread

The Durutti Column - At First Sight
Dmitri Shostakovich - Op. 87 Prelude & Fugue no.4 in E minor
Tim Buckley - Song to the Siren, Phantasmagoria in Two
The Cryin' Shames - Please Stay
Low - Point of Disgust, Murderer
Willis Earl Beal - Evening's Kiss
Rome - We Who Fell in Love With the Sea
Sparklehorse - Apple Bed
Tom Waits - I'm Still Here
The Antlers - Wake

New Disaster by Elliott Smith, I think. Although I don't think it's depressing, it sure it melancholy!

John Williams - Anakin's Fall to the Dark Side | Suite - London Symphony Orchestra

I would suggest Susi Varming´s album "Barefoot in the snow," that should do the job. "Who sent the roses" for instance, "Deeper", or "Wiser now".

Just google Susi Varming, and find links to iTunes etc from her website.

Wishing you all the best!