This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / The Decemberists
"On The Bus Mall". Soaring acoustic guitars + story about teenage runaways in love turning to underage prostitution to survive = utter uselessness on many a person's part.
That goddamn chorus. "But here in our hollow, we fuse like a family/but I will not mourn for you."
Three words: "But oh, Valencia..."
From the same album, "You'll Not Feel The Drowning" and "The Crane Wife 1 & 2" can be quite depressing.
There's a bend in the wind and it rakes in my heart. There's blood on the thread and it rakes in my heart!
"After the Bombs." Two lovers imagining life after war's end.
A whole crapload of songs that Colin Meloy has written can make people cry, most notably (in chronological order) "Odalisque", "Grace Cathedral Hill", "Red Right Ankle", "Eli, The Barrow Boy", "The Bagman's Gambit", especially the lines
"And at the gate of the embassy Our hands met through the bars As your whisper stilled my heart No, they'll never catch me now"
Of Angels and Angles, so damn much. Often interpreted to be about two lovers killing themselves by drowning, it will make you tear up, guaranteed.
"Sons and Daughters" can make people cry bittersweet tears, as it's about a family escaping a war torn country.
Now from The Hazards of Love, we have the last two songs, where Margaret and William knowingly go into the river that will drown them.
"My Name is Leslie Anne Levine, and I've got no one left to mourn for me..." can really hit some people where it hurts.
"The Hazards of Love 4 (The Drowned)" is another one. Especially these lines:
"And as the waves came crashing down He closed his eyes and softly kissed her"
"The Engine Driver" yet? This line, oh my god:
"And I've written pages upon pages trying to rid you from my bones."
The whole song is a tearjerker, (in a more surreal and beautiful way than most of the other examples here.) because the passion and the sadness of the 3 narrators comes across so clearly:
"I'm an engine driver, on a long run, on a long run
Would she were beside me, she's a long one, such a long one."
The first verse sounds like it's about the Engine Driver missing his wife/girlfriend while on the job. On first hearing the song, though, I interpreted it as the engine driver, who has retired, missing the train or truck or whatever he drives, since it would be so odd to can a real woman "A long one."
I perceived this as the Driver missing his love, but also being so isolated from everyone that he no longer differentiates between characteristics desirable for people and characteristics desirable for objects. So his love becomes "a long one", because that's the kind of route he likes best - he has no other frame of reference to draw upon.
"I'm a moneylender, I have fortunes upon fortunes.
Take my hand for tender, I am tortured, every tortured."
Again, there are two ways to interpret this: The sweet way, where the moneylender considers his fortune worthless compared to someone's love (I.e., hand in marriage.) Or, the deals that made his fortune, sealed with a handshake, may have been less than savory, and he's tortured by the memory.
"Yankee Bayonet (I Will Be Home Then)", about the American Civil War soldier and the woman who knows he won't be coming home alive, is another one.
Then there is "The Crane Wife 3".
"And I will hang my head low..."
"Sleepless", released on an album sold to promote AIDS awareness and possible to interpret as being about someone dying of AIDS. Even if that's not what it's meant to be about, it's heartrending.
"I Was Meant For The Stage" is sad enough, but try listening to it after losing an actor friend to suicide. *sniff*
From the latest album: "Please Avery... come home...".
"Rox in the Box" too:
"And it's one-two-three on the wrong side of the lea What were you meant for, what were you meant for? And it's seven-eight-nine, you get your shovel back in line And if you ever make it to ten you won't make it again."
Rise to Me, a song about Colin Meloy's autistic son Henry, is both Tear Jerker and Heartwarming.
"Hey Henry, can you hear me? Let me see those eyes. This distance, between us. Can seem a mountain size."
Listen to the 13th track, the one with the title drop, on What a Terrible World, What a Beautiful World, and you'll hear the narrator simultaneously grateful for the small, everyday joys in his own life and somberly reflecting on a faraway tragedy. Then realize that the title is "12/17/12," and remember what happened on that date...