The 300 series is not well supported. Most people don't even know that 307 exists - they always use 302.
Counterexample: 401 pops up an authentication dialog.
They don't support PUT or DELETE either, which means that people building RESTful APIs have to fudge it. Ugly.
Recent-ish WebKit and Firefox browsers do support PUT and DELETE in XHR requests. I'm pretty sure that any IE < 9 does not, not sure about IE 9 or Opera (though I'd guess Opera does support those verbs).
one major problem is HTML forms doesn't have a way to generate these.
I could invent one pretty easily. Instead of <form action="post">, it could be <form action="put">.