Unreliable Narrator/Music

Examples of Unreliable Narrators in Music include:

Most of the Barenaked Ladies song "The Old Apartment" is meant to imply that the narrator has broken into his ex-girlfriend's apartment in a fit of creepy stalkerishness. Toward the end of the song, he reveals that he and the girlfriend are still together, and have just moved to a nicer house; he's broken into their old place in a fit of creepy nostalgia.
The protagonist of King Diamond's concept album "The Graveyard" claims that he was thrown into a mental hospital because he threatened to expose a politician as a child molester. Since the entire album is from his point of view, and he's an insane killer, it's not clear if he's telling the truth or just crazy.
The refrain of Gaelic Storm's "Johnny Tarr" goes: "Even if you saw it yourself you wouldn't believe it/But I wouldn't trust a person like me if I were you/Sure I wasn't there - I swear I have an alibi/I heard it from a man who knows a fella who swears it's true". The story told in the song is borderline fantasy, wherein the title character dies of thirst in the middle of a drinking contest.
They Might Be Giants do this so much they considered calling one of their albums Unreliable Narrator. To cite one example, "Purple Toupee" is built around the narrator's horribly mangled memories of newsworthy events of the 60s ("I remember the book depository where they crowned the king of Cuba"..."Martin X was mad when they outlawed bell bottoms").
Denton, TX based Slobberbone's "Billy Pritchard" features a father telling his daughter how he doesn't approve of her relationship with a boy in her town, and implies that he killed her brother. Near the end of the song, we learn that the father shot his own son in the back of the head after mistaking him for Billy, and that most of what he had said was a lie.
Eminem played with this for the majority of his career. His 'Slim Shady' character was an obvious parody of the excesses of the gangsta rapper archetype, but a lot of the devices Eminem used with Slim Shady were kept on even after he abandoned the character. How much of Eminem's rapping reflected his own attitudes is a very debatable question.
Rael, the protagonist of the Genesis Concept Album The Lamb Lies Down on Broadway, is practically made of this trope.
In Joanna Newsom's song "Colleen", the story is told by a young mermaid or sea nymph who lost her memory and was subsequently adopted by humans. It's implied that by the end of the song, she's still unaware that she's not human, although it's obvious from the lyrics.

I'll tell it as I best know how, and that's the way it was told to me: I must have once been a thief or a whore, then surely was thrown overboard, where, they say, I came this way from the deep blue sea...

Pink Floyd's The Wall, the movie in particular.
Ludo's song "Lake Pontchartrain" is told from the perspective of a young man who supposedly witnessed his friends' watery, supernatural deaths. But at the last verse he adds; "That's how it happened/Why would I lie?/There were no bodies/I got none to hide", implying that he's being tried for killing them.
Gorillaz bassist Murdoc is notorious for this. He may be the only speaking witness to Noodle's disappearance and apparent death, but he changes the story every time he tells it. Sort of justified in that he claims to be withholding information in hopes of a movie deal. Of course, Murdoc's been known for exaggerating stories and flat out lying on important topics, so it's possible that he's just making things up as he goes.
Pick a Randy Newman song. Any Randy Newman song.
In an interview with Vulture.com, "Weird Al" Yankovic noted that many of his songs feature this trope.

Most of my songs are the Randy Newman-esque, untrustworthy narrator kind of things.

"Margaritaville", by Jimmy Buffett. Just whose fault is it, anyway?

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