Problems as of Konqueror 4.4.1;

•Poor Flash support.
•Poor Java support.
•Poor GMail, Hotmail & Yahoo Mail support.
•Rendering issues.
•Bizarre smearing on certain pages.
•'Load plugins on demand only' causes cursor to stop functioning.
•Serious issues with certain forum software.
•Back arrow will sometimes gray out randomly.
•No unified option to quickly clear browser data (cookies,cache,history,etc).
•Overall stability is still a serious issue.

Apart from the 'Load plugins' & graying out back arrow, these issues have been prevalent since 4.0. That was over 2 years ago.

GMail works good if you use fake identity of browser. I found that Opera 9.62 works the best.
But other problems are true. Well flash mostly works for me, YouTbe works, and other video-players mostly work (I think the one that's used on failblog doesn't)...
But ceratin sites that use flash for non-video purposes don't work.
Smearing is also happening to me.
I think KDE devs are aware of this, and if you've read few blogs on planetKDE there are few ideas of making rekonq default browser for KDE, but it's just an idea, and it won't be implemented soon. Even rekonq devs were against making current version of rekonq the default browser of KDE.

I have to say that Konqueror is my default browser and I tried quite few browser even Google Chrome, Opera and Friefox, but I prefer Konqueror over all of them.

konqueror user here

i like it, expecially on blip.tv and ogv video (kmplayer_part) instead of flash...

but yea it would be nicer if it will be feed better.... but.... try to use webkit engine on konqueror instead of khtml...