Two of the applications that I like to use are Konqueror and Gmail. Konqueror is the best in performance and memory consumption on KDE, and I like to use it for casual browsing. Gmail is probably known to everyone today, not only because of its UI but also because of its good spam filters, productivity and things you can do with it.
Unfortunately these two are not compatible with each other, probably because Konqueror is very good in compliance to standards, but not good with handling deviations from it. There is a way of cloaking your Konqueror as some other browser by changing the browser identification. But even the Firefox cloak does not work.
One solution I found was to use nocheckbrowser argument which tells Gmail to load its standard interface without worrying about browser compatibility. As of today it seems to be working in Konqueror, but I am not sure if this is intentional or a coincidence. Konqueror is definitely not part of the supported browser list yet.
