When to log a stacktrace for a caught exception

Question

I recently asked whether to report the getMessage() text of a caught exception. Rather surprisingly, most answers misunderstood my question and thought I was asking whether

Answer 1

I usually report exceptions not related to GUI. GUI Exceptions are common, and in heavy Swing programs usually happens.

Generally speaking, i'm deeply interested in: DB related exceptions, My-own-stupid-errors exceptions (array out of bounds, and stuff like that) and some other things that cannot be handled by me, like WebServices etc.

I think it depends the type of system, if it's a public one (like a website), or if it's private (intranet sites, GUI local)