Each of the three have aspects I'm interested in, along with giant cans of worms I don't care about for the present discussion.
REST - Representational State Transfer
The part I care about: Each request is mostly-stateless (I except authentication). A URL refers to a noun -- a thing (or list of things) you are viewing or manipulating.
The part I care about: Fetch stuff from the server and display the result in the current page without going to another page. Change the minor-state but not the major-state of the page.
The part I care about: Re-Inversion of control... make it so that the server-side is an ongoing processess instead of one that terminates between each request. Effectively stop programming state machines though explicit lists of states and transfers. Instead use the language's built-in control structures.
I want simple, memorable, URLs which reflect the current state of the application and a linkable jump-point. I'd like for the back button to go back to the most recent linkable spot.
I used to try to make every link on the page secretly be a button. That way when you navigated away from the page the webapp would get the chance to say "wait a second! Are you sure?" if it so desired. But thats a horrible way to do it -- well, if nothing else because its ugly.
But the point is that I want that functionality; if someone is in the middle of a complex process but then navigates to the front-page I want to make sure they get the chance to save their work (or at least that their work is transparently saved).