No, don't catch everything. Exceptions propagate higher up on the stack. All you have to do is make sure that the exception is caught before it gets to the top of the stack.
This means, for instance, that you should surround the code of an event handler with a try/catch block. An event handler may be the "top of the stack". Same for a ThreadStart handler or a callback from an asynchronous method.
You also want to catch exceptions on layer boundaries, though in that case, you might just want to wrap the exception in a layer-specific exception.
In the case of ASP.NET, you may decide to allow ASP.NET Health Monitoring to log the exception for you.
But you certainly don't ever need to catch exceptions in every method. That's a major anti-pattern. I would loudly object to you checking in code with that kind of exception handling.
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