I personally use the stock browser because it's so smooth and the text rearranges to fit in the screen, but I much prefer Chromes tab switching. Also the stock browser has crashed on me twice :-(
I use Chrome as well as Dolphin Browser. The latter has recently become better than the former with the V10 update. However I use Chrome a lot because of bookmark and current tabs syncing between my device and my computers.
Chrome because I like it a lot. Also I use it on more then 1 device and like how bookmarks and recently open sites/sites open on other devices are shared, for example.
I like the stock browser because it is smarter about resizing pages when you double-tap to zoom in/out. That's very important to me because it makes a significant difference on several sites I visit frequently. It also seems a bit better about consistently delivering pages in desktop mode when you request it. It also still has flash support for those increasingly-rare times I need it. In the end it's the resizing behavior that most matters to me.
I tend to use Chrome Beta instead of regular Chrome or the other browsers. It runs a bit smoother than regular Chrome and has some fixes that aren't available for it at the moment. Plus, I get to have all of my bookmarks and such synced from my other devices. It's really up to you; I recommend trying them out and then deciding from there.
By the way, Chrome Beta can't be found on the Play Store app; you have to visit the Play Store on its online site (play.google) and have it downloaded to your phone from there.
I left the defaults free for weeks after buying the device, I would simply pick Stock or Chrome 'just once' randomly and see how I got on. After a month of doing this, I've eventually set the Stock as default. It wipes the floor with Chrome imo. There's just something I can't put my finger on with Chrome that I dislike, it's the same on the Nexus 7. I love the way the tabs work on Phone Chrome but other than that, stock browser all the way.