So what happens if a user agent (browser) uses HTTP/2 to request a webpage on shared hosting currently?
The short version is that the client is responsible for determining the protocol when it connects to the server. It will try connecting via HTTP/1.1 and auto-negotiate connectivity via HTTP/2. HTTPS is a little different, but it should still be automatic if the client and server both support the new protocol. More details here - http://http2.github.io/http2-spec/index.html#discover-http
Thanks, good info
Here a testing tool to see if a site supports HTTP/2: https://tools.keycdn.com/http2-test
So far, sites on shared hosting still do not.
Using this tool:
It seems even the Dreamhost site isn't HTTP/2
It is not correct to say that DreamHost doesn't support HTTP/2. If you switch your VPS, DreamPress or dedicated to nginx, you'll have HTTP/2 on your shared hosting domain.
Justin Lund has already answered this topic sharing the rough roadmap for a wider deployment of HTTP/2. I think we can close this thread now and until there is an official announcement from DreamHost.