If you're targeting Mobile Safari, it's comparatively easy to see your error messages when you're debugging, just enable it in the settings. It gets tricky with a UIWebView though, and we ended up using this custom URL scheme hack (which requires some native code changes) to get log messages appearing in the device console. It's also worth knowing that you can view the console even when you didn't run the app through the debugger (for example if you've installed it through the app store) by plugging in and looking in Organizer->Devices. You can even buy apps to let you view the console natively, which should make you think twice about putting any private information you don't want other apps to access in log messages!
You should check out the new iOS 6 remote debugger, which works with both Safari and UIWebView code. It's been extremely useful for digging into CSS issues, and saved our bacon when tracking down some weird script loading problems.
Catching errors in the wild
The most challenging part is getting information on problems that are happening to users with the released app. If you can't reproduce the issue locally with a device plugged in, how can you tell what went wrong?
The first step is attaching a callback to window.onerror(), which will be called whenever there's an uncaught exception. In iOS 5, you only get the error message, not the file or line, and for various reasons we've had to minify and inline code anyway, so iOS 5's addition of the line number and file name isn't very helpful. What we really need is the call stack, which just doesn't get returned in any form on Mobile Safari.
This won't run out of the box, but it should give you an idea of what we're doing. As part of our server-side code we have an error-reporting endpoint that we post the details of any release errors to, /jserror, and that sends on an email to the team.
The heavy lifting happens in the wrapFunctions() call, which replaces each function in an object with a wrapper that first calls the supplied 'before' function (in our case just pushing onto the callstack), then the original function, followed by 'after'. There are no guarantees about the correctness of the code in all cases, the prototype stuff especially scares me, but it has worked in practice on our code base.
I tend to use this pretty sparingly to wrap our own code, rather than jQuery or other frameworks, since most of the errors are in our functions, and I'm worried about sprinkling too much voodoo over our code base. Despite those caveats, it's been a massive help in tracking down our issues.