Argthtjtr

Yes. But it's so-so bad.

https://en.wikipedia.org/wiki/The_Return_of_Captain_Invincible

The Return of Captain Invincible is a 1983 Australian musical comedy superhero film starring Alan Arkin and Christopher Lee. It grossed a mere $55,110 at the Australian box office despite a budget of $7 million.[2]

There's a Youtube review

Not a movie, but a play and then a TV special. From Wikipedia:

It's a Bird... It's a Plane... It's Superman is a musical composed by Charles Strouse, with lyrics by Lee Adams and book by David Newman and Robert Benton. It is based on the comic book character Superman created by Jerry Siegel and Joe Shuster and published by DC Comics.

1966 for the play and about a decade later for the TV adaption.

(It's always been one of those things I am intensely curious but also terrified of ever seeing)

There is in fact a Bollywood movie musical about a superhero!

Mr. India

A poor but big-hearted man takes orphans into his home. After discovering his scientist father's invisibility device, he rises to the occasion and fights to save his children and all of India from the clutches of a megalomaniac.

I would actually recommend watching it. It's funny/corny like Dr. Horrible but not too corny (I hate corny). It's actually on YouTube.

YouTube - Mr. India

You could say Dr. Horrible's Singalong Blog might be considered a superhero and musical. Though it is a parody (with some good songs), it may not be what you have in mind.

Depending on where you draw the lines for "superhero movie" and "musical", various of the Lego Movies might count. For instance, here's a musical number from Lego Batman.

Not a movie, but theatre play (whole on youtube).

It's called "Holy Musical B@man!" by Team StarKid (also did the Very Potter Musical)

Youtube - Holy Musical B@man!

There's Disney's animated musical film Hercules. The protagonist wasn't (until the end, at least) a mortal man, but had superhuman strength, and spent a lot of the flick training with a professional "hero trainer" Philoctetes, and in musical bits.

It wasn't billed as a "superhero" movie at the time, but in the 20th century that didn't exactly exist as a defined genre yet, and being a "Disney movie", complete with musical numbers, was probably considered a bigger sales driver.

His origin story in the movie is very similar to Wonder Woman's in the 2017 movie. Both got their powers through being children of Zeus, but were sent to Earth to live as mortals. (Arguably, the original Thor movie had a similar setup as well).

The movie's theatrical release was in 1997.

Depending on whether you consider Buffy to be a superhero, and a TV episode to be a movie, possibly Once More with Feeling.

The Hungarian-Canadian cartoon Cat City from 1986 could qualify, as it features many musical inserts and the protagonist has abilities that are, while not strictly superpowers, definitely above those of the normal mice: folding his tail to the shape of a key, cutting a ship's rope with his teeth, using his tail as a propeller to cross an ocean, bending an arm-wide metal pipe into U shape, memorizing multiple pages of technical drawings in seconds, etc.