Radiohead/Tear Jerker

Radiohead is weird, disturbing... yet has a large assortment of these type of songs.

"Exit Music (For a Film)". It was written for a Romeo and Juliet adaptation, so the Tear Jerker quality is imported from a much older source.
"How to Disappear Completely" is even worse.
Some people can feel melancholy from listening to "Karma Police".
"Fake Plastic Trees", "Street Spirit (Fade Out)", and "Let Down".
- The lines "One day I'm going to grow wings / A chemical reaction / Hysterical and useless" from "Let Down" deserve special mention.
- The former is even more of a Tear Jerker in Brazil, where it was the soundtrack for a PSA on Down Syndrome.
- There's a lot of great lines in "Fake Plastic Trees" but especially heartrending is the final verse: "she looks like the real thing/she tastes like the real thing/my fake plastic love /[...]and it wears me out/it wears me out."
"You and Whose Army" stands out especially, although it might be closer to Awesome Music -- but Your Mileage May Vary.
"No Surprises" has sent many people into multi-day cascades of melancholy when they actually tried to reconcile the dissonance between the music and the lyrics.
- The little moment of peace at the end ("Such a pretty house, such a pretty garden") just increases the Tearjerkiness.
- "Motion Picture Soundtrack", from the end of Kid A, has a similar air of desperation to it.
"Harry Patch (In Memory Of)" was written as a tribute to Harry Patch, the last surviving soldier to have fought in the trenches during World War I, who has recently passed away.
"All I Need" is definitely incredibly melancholy and apathetic.
"Creep". Never mind that it might be cliche, and Radiohead hates the damn song. Some people still find it impossible to not cry at the song..
- While Radiohead's "Creep" is sad in and of itself, this cover by a homeless man can tug at the heartstrings of even the most resolute of individuals.
"Lucky" in itself has a somber, heart-rending effect, but when set to a war child video, the song takes an entirely new light.
"Videotape". Come on. It's a suicide note in the form of a song.

"No matter what happens now / I won't be afraid / Because I know / Today has been the most perfect day I've ever seen."

"Talk Show Host" can often leave some people feeling depressed.
"Go Slowly" and "Last Flowers to the Hospital", off of the extra In Rainbows disc, are both incredibly melancholy and beautiful.
- Then there is "Four Minute Warning".
"Bulletproof... I Wish I Was" can usually tug on one's heartstrings.
- Seriously, don't listen to that song in a depressive mood. Its just... overkill. That song is extremely sad.
"Lewis (Mistreated) has a very catchy and upbeat riff, but the depressing lyrics convey a different message. One of the lyrics is "don't do it, don't, jump." Thom Yorke even sounds like he's crying (or at least very distressed) when he sings this.
- It doesn't help that Thom pauses right before he says "Jump", which might imply he did jump.
"Fitter Happier" never fails to bring certain people to tears. The line "Baby smiling in back seat" can be especially triggering.
The band's music tends to get used in heartbreaking Public Service Announcements like this and this
The beginning of "Cuttooth." It's not so much the words but the longing way Thom sings it.
The live version of "Fog." "Where did you go bad...? Did you go bad...?" *sniff*
Oh sweet God, "Pyramid Song." "...and we all went to heaven in a little row boat | there was nothing to fear, nothing to doubt..."
- In the "Live in Praha Concert", Thom dedicates it to Franz Kafka before they play it. The live version featuring Jonny's bowing of his electric guitar makes it all the more haunting. ;~;.
- The video just about matches the depressing lyrics. Computer animated, we're introduced to a man standing on the roof of a skyscraper in the middle of an ocean, revealing he's the last survivor of a flood that destroyed his city, and possibly the rest of the world. Suitng up, he dives into the water and observes the city's sunken ruins. In the end, he swims into what is apparently his family's house, takes a seat, and unplugs his oxygen cord so he can rejoin his loved ones. Quite possibly the most amount of emotion you'll ever feel for a CGI stick figure.
The Tourist. "Hey man, slow down... slow down..."
"A Wolf at the Door". While it's not as slow and melancholy as most of the songs on this list, it is sung with incredible bitterness and desperation. You can't help but feel the pain in Thom Yorke's voice.

"Let me back / Let me back / I promise to be good / Don't look in the mirror at the face you don't recognize / Help me, call the doctor, put me inside"

"Give Up the Ghost," off of The King of Limbs, is rather tearjerky.
- The version from the In the Basement video album outdoes the album version's tearjerkiness by a mile.
"Codex". Pretty self-explanatory as to why.

"Sleight of hand / Jump off the end / Into a clear lake / No one around / Just dragonflies / Flying to the side / No one gets hurt / You've done nothing wrong / Slide your hand / Jump off the end / The water's clear / And innocent / The water's clear / And innocent"

"True Love Waits." Oh god.

"I'm not living/I'm just killing time..."

There There. Thom just sounds pained at times. And was allegedly brought to tears after hearing the final mastered version of the song.

"We are accidents, Waiting waiting to happen."

RAAAAAAAAAAAAAAIN DOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOWN.... RAAAAAAAAAIIIN DOOOOOOOOWN, COME ON RAAAAAAAAAAAIIIIIIN DOOOOOOOOOOOOOWN...... ON MEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE...................
"Let Down". Just..."Let Down". As if the subject matter isn't melancholy enough, this troper considers the "You know where you are..." bit to be one of the most heart-breakingly beautiful moments in music history.
"I Will".
The backing vocals to "(Nice Dream)":

"If you think that you're strong enough./If you think you belong enough."

We hope... that you choke... that you choke...