I think you can use any js-based ssb-server at the moment. I mostly use Patchwork. I'm keen on adding support for ssb-browser-core but I don't know how.
The JS based
scuttle-shell should work as well.
If I get the funding, I'll be able to put work on supporting go-ssb. Once I learn how to do multiple "platforms" then the path to add support for
ssb-browser-core should become clearer.