Looking around, I can\'t name a single web application (not web service) that uses anything besides GET and POST requests. Is there a specific reason for this? Do some browsers
This depends on your browser and Ajax library. For example jQuery supports all HTTP methods even though the browser may not. See for example the jQuery "ajax" documentation on the "type" attribute.
The Restlet Java framework lets you tunnel PUT and DELETE requests through HTML POST operations. To do this, you just add method=put or method=delete to your URI's query string, eg:
http://www.example.com/user=xyz?method=delete ...
This is the same as Ruby on Rails' approach (as described by @ars above).