There really aren't that many horror/suspense musicals, and the ones that exist are mostly dark comedies with horror elements. Think Bat Boy, Heathers, Carrie, Evil Dead, Silence!, Toxic Avenger, Little Shop of Horrors, Dance of the Vampire, Young Frankenstein, Shockheaded Peter, Re-Animator, and Zombie Prom.
There are some straight up horror/suspense/gothic musicals that are totally hit or miss depending on the production. I'm a big fan of Nevermore, but it's a pretty bleak Edgar Allan Poe adaptation. There have been numerous Frankensteins and Draculas. Jekyll and Hyde and Lestat exist, as well. Jane Eyre dabbles with some good old gothic suspense, but is more romantic than not. Phantom of the Opera does the same (only much better).
I'm a big fan of the unlicensed parody musical A Shoggoth on the Roof, which is a Lovecraftian spin on Fiddler on the Roof. Recordings exist, but the Fiddler rights holders shut down any real attempt at a production since Shoggoth uses all the original orchestrations and only changes lyrics. Apparently, someone wrote a brand new score for it at some point but it never got any traction. The genius is how well the existing music works for a tale of absurd horror. This documentary features a lot of the lyrics and some backstory: https://www.youtube.com/watch?v=2bHqURZrgIc