Depressing songs? That's my expertise. I'll try to keep it short:. Off the top of my head:

Richard Ashcroft - The Drugs Don't Work - Lyric: "'Cause baby, ooh, if heaven calls
I'm coming too, just like you said. You leave my life, I'm better off dead"

Kate Bush - This Woman's Work - about a woman (almost) dying during childbirth. Think most people can relate to the thought of their mother (or mother of their children) dying. (And if that's not you, internet hugs. Sorry, if that's all I can offer you.)

Soft Cell - Say Hello, Wave Goodbye - Mark Almond isn't the best singer, this sounds like a cheesy 80s song about a break-up.... and then the chorus hits. Anger and bitterness seem to mask true pain.

Regina Spektor - Becoming All Alone Again - Key lyrics: "Why doesn't it get better with time?
I'm becoming all alone again."

The Knife - Is it still light outside? - a cheery song about the doctor coming for a visit after something happened.

Bowie and Mercury = Under Pressure - Aftersun version - Key lyric: "Why can't we give love one more chance? Why can't we give ourselves one more chance?" and "this is our last dance" - in this version Bowie never gets to finish the lyric. Love doesn't get one more chance. The movie's equally depressing.

Pink Floyd - High Hopes - about regret and growing old. Key lyric: "Running before time took our dreams away, leaving the myriad small creatures trying to tie us to the ground to a life consumed by slow decay". Obviously Pink Floyd has other great songs, I chose one of them.