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Gorillaz's Murdoc rapidly grew out of Woobie status of any kind, but in his preteen days, he definitely fell into this category. According to Rise of the Ogre, he was a horrible child, but given that he was a victim of Financial Abuse and public humiliation by his father, suffered regular beatings from his brother, and claims to have lost his virginity at the age of nine, and it's horribly possible that he's not making that up, you can't help but pity him. His pitiable status resurfaces a little in his Pirate Radio broadcasts, during which he's perpetually drunk and rambling about how lonely he is.
Donato from Gian-Carlo Menotti's opera, Maria Golovin. He's a young man, blind from war, living with his mother and building birdcages. He's very lonely and he falls in love with a woman who lodges with them. Then he becomes fearful and jealous and has outbursts of anger, culminating in him grabbing a gun with the intent of shooting her.
Eminem's stage persona definitely comes off as one of these, and the man himself has not had an easy life. Sure, he threatens to kill his wife repeatedly, but there's no doubt that his life has sucked.
Likewise Tyler, The Creator's stage persona, and when performing as "Wolf Haley" or "Samuel" he brings it to Woobie, Destroyer of Worlds. Sometimes the man himself can be this, due to his obnoxious behaviour but somewhat serious Artist Disillusionment and genuine emotion in his music.
Are they a character in the Evillious Chronicles? Do they have a name? Congratulations, they're either this, Woobie, Destroyer of Worlds, or The Woobie.
Pink from The Wall beats his wife and essentially becomes a Nazi-wannabe, but songs like "Bring the Boys Back Home" and "Hey You" remind us that there is a decent, redeemable person underneath all of that.
The subject of the Country Music standard "Life Turned Her That Way".