We're all calling these "hypermedia APIs" these days.
> with an associated client that actually uses it?
I have written a toy client here: https://gist.github.com/steveklabnik/2187514
You can run it against this site, written in node: http://alps-microblog.herokuapp.com/
Or this site, written in Rails: https://rstat.us/
It (should, I haven't tried it in a long while) work with both just fine. They both use the ALPS microblogging spec. yay generic clients!
As for people who have 'more real' ones: GitHub, Twilio (partially, more in the future), Balanced Payments (YC W11, iirc), Comcast (though that's internal :() Netflix has aspects, FoxyCart.
This year will be the year of examples.