Inspired by this thread.
What album (did)do you listen to to cope with hard times? The one that made you forget that the job is stupid and menial, the boss is an ass, and the weather has been absolute crap for the past week and a half. The one guaranteed to defunk my life back when I was in the U.S.A.F. was Lynda Carter’s Portrait(1978). I was at Warehouse Records in Hollywood and someone had left it in a listening booth, so I stuck on the headphones and heard this. I double-checked to make sure it wasn’t a Karen Carpenter album in the wrong sleeve, took it home, and it’s been cheering me up ever since.
As a mopey teenager, it was Joni Mitchell’s Blue.
As a frazzled support technician, when I wasn’t working I couldn’t stand the sound of human voices – even lovely singing ones – so it was John Renbourn’s The Lady and the Unicorn.
More recently, Chin Up, Cheer Up. 
King’s Singers Madrigal History Tour has been my going-to-sleep album for I don’t know how many hundreds of nights running when I’ve been living overseas and having some trouble coping.
Something about the surgically precise diction and pitch combined with the loveliness of tone and charm of musicality just makes that album my walled garden of security and beauty. “Margot, laborez les vignes, vigne vigne vignolet…”
ETA: Oh, and it’s not an album, but the song “Work’s Too Bloody Hard” by the Prodigals has got me through more all-night grading and writing sessions than I would care to count.
Back in about 2001/2002 when I out of the blue developed an anxiety syndrome, complete with panic attacks and all that, and I felt I was slowly going out of my mind (eventually, it passed), it was Sleater-Kinney’s Dig Me Out album, along with Call the Doctor that kept me feeling sane. Whenever I would start feeling a panic attack come on, I would just try to play back a song off one of those albums in my head to distract me, and it would work.
Listening to anything by Kesha never fails to cheer me up. If I have to answer the OP’s question, I’d say her Cannibal EP because of a song called “Crazy Beautiful Life.”
Joe Satriani’s self titled album from 1995
Pink Floyd’s The Wall helped me when I was feeling lonely and trapped in my own home with an overbearing family. “Mother” in particular struck a chord with me.
Quadrophenia.
“You’ll all see I’m the one.”
Amen brother. Amen.
Much underrated album, and I like it that way.
R.E.M.'s Automatic For The People, when I was a shy, underconfident teenager who wasn’t enjoying school. Nowadays, now that I don’t get melancholy anymore, just pissed off, it’s Weight by Rollins Band, or The First Five Years, a best of Black Flag complation.
The original cast recording of The Last Five Years was great for sorting through the complex emotions of the end of a bad long term relationship. This song in particular resonates with me Nobody Needs to Know - The Last 5 Years - YouTube.
Biko. Peter Gabriel.
Nothing captured better.
The Beatles’ “Abbey Road.” Got a copy for my 21st birthday in 1969.
In 2000, in the months immediately after my husband passed away (I was 51), I listened to it over and over and over.
Great album. In my top 5 for sure. Wonderful stuff.