Specific scenario
While I don't want to concentrate on only the issue I am having, here are some details of the current issue we face and how I've tackled it so far.
The issue occurs when the user interacts with the user interface (a TabControl to be exact) at a particular phase of a process. It doesn't always occur and I believe this is because the window of time for the problem to be exhibited is small. My suspicion is that the initialization of a UserControl (we're in .NET, using C#) coincides with a state change event from another area of the application, which leads to a font being disposed. Meanwhile, another control (a Label) tries to draw its string with that font, and hence the crash.
However, actually confirming what leads to the font being disposed has proved difficult. The current fix has been to clone the font so that the drawing label still has a valid font, but this really masks the root problem which is the font being disposed in the first place. Obviously, I'd like to track down the full sequence, but that is proving very difficult and time is short.
Approach
My approach was first to look at the stack trace from our crash reports and examine the Microsoft code using Reflector. Unfortunately, this led to a GDI+ call with little documentation, which only returns a number for the error - .NET turns this into a pretty useless message indicating something is invalid. Great.
From there, I went to look at what call in our code leads to this problem. The stack starts with a message loop, not in our code, but I found a call to Update() in the general area under suspicion and, using instrumentation (traces, etc), we were able to confirm to about 75% certainty that this was the source of the paint message. However, it wasn't the source of the bug - asking the label to paint is no crime.
From there, I looked at each aspect of the paint call that was crashing (DrawString) to see what could be invalid and started to rule each one out until it fell on the disposable items. I then determined which ones we had control over and the font was the only one. So, I took a look at how we handled the font and under what circumstances we disposed it to identify any potential root causes. I was able to come up with a plausible sequence of events that fit the reports from users, and therefore able to code a low risk fix.
Of course, it crossed my mind that the bug was in the framework, but I like to assume we screwed up before passing the blame to Microsoft.
Conclusion
So, that's how I approached one particular example of this kind of problem. As you can see, it's less than ideal, but fits with what many have said.