I use an old version of Firefox (3.0), because I prefer it. Unfortunately some web sites are starting to break on this version though. Currently I use a separate machine with a newer Firefox for the occasions when I need a newer browser. Eventually I might have to use something newer on my main machine though.
The only annoyance I have with 3.0 is that when any browser tab gets stalled for some reason, it freezes up the entirety of all of Firefox, including all tabs and all windows. Maybe newer versions actually have a useful implementation of SMP, but I'll be skeptical until I see it. As a programmer, multi-CPU setups have always been interesting. As a user, the reality of them has always been underwhelming, not just with Firefox.
When watching youtube, I sometimes run out of space on my partition that I use for temporary files. I wish Firefox had an option to assign an overflow folder for large files (.flv's in this case).
I've ignored Chrome as I don't like Google in general, but I may look into it.
Palemoon sounds interesting.
Something I must say I hate about the newest Firefox versions, as well as Thunderbird, is that by default they will automatically update themselves, and any addons you have installed, without asking. It's literally a race to disable that behavior before it's too late. And with the newest versions I think even the disable option is hidden away in that "about:config" thing. Mozilla has developed a paranoid nanny-state attitude that I can't stand. I want a browser that treats me like an adult.
My main complaint with web browsing today is the way web page design has become excessively and frivolously client side scripted, but that's not something I can blame the browsers for, at least not directly.
IE6 was fast, I never agreed with the widely reported claims of Firefox being faster. I even tested this claim using a web page which was making this very argument. On whatever machine I was using at the time, probably a Pentium-II, it scrolled and operated quickly and easily on IE6 but was straining and lagging on Firefox (1.x or 2.x at that time). As my computers got faster it no longer mattered though.
The automatic prompts to install addons for IE6 was incredibly dumb and was the primary cause of it's virus problems. That behavior can be disabled but it's not a convenient process - you have to hack one of the DLLs to get rid of a popup message that will otherwise keep appearing. I think XP SP1 or 2 fixed that.