I've got Opera running off a USB flash drive (256mb) with no problems at all. Program isn't even slowed down. All I had to do was remove the drive letter and : from the ini files for instance M:\Opera\profile\... would become \Opera\profile\...
Opera can understand it without the drive letter.
Now I can use Opera on my college's library computers. No more junked up IE (they got all kinds of junk on these computers).
Plus I can go home and use filesync to synchronize the flash drive opera and the opera on my desktop.
Just need to figure out how to put Aspell on the flash drive so the spell check still works.
From what I see online though Firefox's extensions aren't gonna work on a flash drive (drive letter changes). Can't really compare to Opera's feature set without the extensions.
WARNING: This is an unofficial Firefox package. It is an official build, but has been modified by me. No warranty is expressed or implied.
IMPORTANT NOTE: As most USB drives advertise a finite lifespan of anywhere from 100,000 to 1,000,000 erase cycles, this package has browser history and disk cache disabled to minimize impact to your USB key.