Amused to, by, with: Which one?

“Amused by” is common, and “amused at”, “amused with” are both used.

You sometimes see “She was amused to hear that …” But here “to” isn’t a preposition (it is forming the infinitive of the verb), but otherwise “amused to” isn’t common.

There is another pattern “… to death”. You can say “He was shot to death” or “She was loved to death”. It means “He was shot, and it killed him” (loved to death is a figurative use of the expression). The album title “Amused to death” follows this pattern, but it is deliberately an odd expression. People don’t say “amused to death”. Album titles often take a common expression and change part of it to create something original.