Personally, I would target "any HTTP server that implements the PSGI spec". This means Twiggy, Starman, Corona and HTTP::Server::Simple::PSGI. I would assume that most of the other servers can be made easily into a PSGI-compliant server, or already are.

I think there are adapters to make PSGI work as FCGI so you should be able to let your application run behind any other webserver, like Apache or Nginx.

If you choose one of the other server implementations, you tie your program to its protocol. This is not inherently bad (see HTTP::Server::Simple), but you limit your options when combining your program with another webserver. For automated self-testing of the program, especially if it has Javascript components, having a stand-alone webserver is invaluable.