Eels/Tear Jerker

This alternative rock band has so many incredibly sad songs. Songs ranging from his sister, his mom, his father, himself, lost love, death, sickness and a whole lot of things. Some are even sung along really catchy and poppy tunes, however there are a lot of songs that are slow and depressive.

Pretty much all of the album Electroshock Blues, especially "Dead of Winter" and the final track "P.S. You Rock My World" ("...everyone is dying, and maybe it's time to live"). The whole album may be THE most depressive album ever, and is his best work.
Also especially "Last Stop: This Town". It helps if you know beforehand that it's about the suicide of the songwriter's sister.

"What if I was not your only friend in this world? Will you take me where you're going if you're never comin' back?"

"Somebody Loves You" is another one.
Then there is "I Need Some Sleep", an extremely sad song.
"Things The Grandchildren Should Know" is another big one. The song can hit really hard, in particular this line:

"I do some stupid things, but my heart's in the right place, and this I know."

"Flower", "Manchild", "Climbing to the Moon", "Railroad Man", "If You See Natalie", "I'm Going to Stop Pretending that I Didn't Break Your Heart", "A Line in the Dirt", "Little Bird"... the list goes on and on.