MP3, OGG (and whatever else your browser supports natively), MOD, XM, S3M, IT (and all other formats implemented by libopenmpt), SID, SAP (and other Atari 8-bit formats implemented by ASAP), SNDH, PSG, VTX, STC, SQT and PT3 formats.
Getting every last detail and edge case of demoscene music formats right is hard, and so Cowbell is built on top of existing tried-and-tested playback engines. For tracker music formats, we use an Emscripten build of libopenmpt, the most mature and comprehensively-tested module player library available. Players for ZX Spectrum formats are either a direct translation of the original Z80 player routines, or the Z80 player routines themselves running under emulation.
Requires CoffeeScript, Pasmo http://pasmo.speccy.org, Closure Compiler, Perl and Make. Or just download the latest release.
You have to write your own HTML but the doc/usage.md file describes it fairly well. Or you can grab the HTML page from the demo site and use that, probably.
Stefan Ossowskis Schatztruhe has just uploaded a four disk archive of MODs, S3Ms, XMs, and other tracker music files to archive.org. If you're into chiptunes (Amiga MODs, in particular) this is at least a year's worth of music to listen to.