I personally use Firebug/Firebug Lite and on IE let Visual Studio do the debugging. None of these do any good when a visitor is using some insane browser though. You really need to get your client side javascript to log its errors to your server. Take a look at the power point presentation I've linked to below. It has some pretty neat ideas on how to get your javascript to log stuff on your server.
Basically, you hook window.onerror and your try {} catch(){} blocks with a function that makes a request back to your server with useful debug info.
I've just implemented such a process on my own web application. I've got every catch(){} block calling a function that sends a JSON encoded message back to the server, which in turn uses my existing logging infrastructure (in my case log4perl). The presentation I'm linking to also suggests loading an image in your javascript in including the errors as part of the GET request. The only problem is if you want to include stack traces (which IE doesn't generate for you at all), the request will be too large.
Tracking ClientSide Errors, by Eric Pascarello
PS: I wanted to add that I dont think it is a good idea to use any kind of library like jQuery for "hardcore" logging because maybe the reason for the error you are logging is jQuery or Firebug Lite! Maybe the error is that the browser (cough IE6) did some crazy loading order and is throwing some kind of Null Reference error because it was too stupid to load the library correctly.
In my instance, I made sure all my javascript log code is in the <head> and not pulled in as a .js file. This way, I can be reasonably sure that no matter what kinds of curve balls the browser throws, odds are good I am able to log it.
