I just use Safari. Unless there's a feature you really want/need that Safari does not have, the loss of system integration is bothersome.
For those occations that I do need a special feature I use iCab Mobile. It's packed with features. So much so that they are actually a bit of a pain to use, because you've got so many settings and menues to sort through. But as a second browser, it covers almost all the nitche needs.
The other browser I keep is Puffin. Puffin is a Flash enabled browser. Because of how it works (translating the websites on a remote server) it often works with other pluggins and extentrions as well: Silverlight, Java, etc. I haven't actually needed it for a couple of years now, but it's nice to know it's there if/when I do.
I also have Chrome; though I've never run accorss a site that worked any better in Chrome than Safari, or a feature in Chrome that I felt worth the trouble of switching back and forth between browsers.