<p>When should you NOT use a singleton class although it might be very tempting to do so? It would be very nice if we had a list of most common instances of 'singletonitis' that we should take care to avoid.</p>

I know many have answered with "when you have more than one", etc.
Since the original poster wanted a list of cases when you shouldn't use Singletons (rather than the top reason), I'll chime in with:
Whenever you're using it because you're not allowed to use a global!
The number of times I've had a junior engineer who has used a Singleton because they knew that I didn't accept globals in code-reviews.  They often seem shocked when I point out that all they did was replace a global with a Singleton pattern and they still just have a global!