I just got my first smart phone 2 weeks ago and I started out with Opera Mobile. I picked it over Opera Mini because Mobile allowed me to view the sites as you see them on your PC. I would have stuck with it and never tried another browser if not for my girl complaining it wasn't allowing her to do everything she needed to do for her schooling with it. So I went on the hunt and tried firefox and it failed. Tried Skyfire too and still wasn't cutting it. I then installed Dolphin Browser HD and finally found a browser that does everything she needed a browser to do. After a few days of using it for my browser as well, I have to say I think I found the right browser for me.
So what browser do you use on your smart phone?