I run Firefox or Mozilla on my Linux boxes and am able to view well over 95 percent of the sites I visit. I use Firefox for all of my ecommerce stuff incuding bill paying and have been very happy with the results. Even on Windows, I have IE, Firefox and Opera all there, but I won't use IE, because of the swiss cheese like security, unless a site that I badly want to see will do nothing except IE.
My favorite right now is Firefox. It is fast, reliable, and doesn't have as many security problems as IE, and it runs under both Windows and Linux, although I run Linux most of the time.
The only thing I can't access with Firefox is that Toshiba copier. (Kinda odd when you think about it. The "brains" of that copier is a Linux system.)