TBC by Tin - Just starting...
Library Based
Simple libraries and provides easy integration and flexibility.
jQuery - Has become the de-facto standard, but I am also looking for a commercial option for web applications.
Sencha - HTML 5 Platform with tools to visually build for mobile and desktop web applications. Buy license.
YUI - Tried before in 2010, and found it to be quite slow.
Integrated
In general Integrated UI frameworks are more intrusive and require tight integration and sometimes dictate how your application actually flows.