Is it possible to display the current state of modifier keys? I have a keyboard which is a little flaky, and occasionally one of the modifier keys becomes stuck. Realizing I’ve entered this state often takes a while. Is it possible to display — e.g., in the menu bar or in the style of Sticky Keys — which modifier keys are currently down?

I don’t need the Sticky Keys functionality, so turning that on wouldn’t seem to be a solution.
There is one native solution I can think of, and that's to show the Keyboard Layout window. This can be accessed through the input menu in the menu bar*, and will show you all keys and their current states. A bonus to this for normal circumstances is that you can see where option and shift-option special characters are mapped.

Another possible solution involves a third-party app such as Keyboard Maestro. If you create a macro triggered by the press of a modifier key, you can tell it to, say, display a Growl notification with a relevant message. I'm not exactly sure how you'd want to do this, or what exactly you'd want it to do in the first place, but many of these programs have tons of options.

*To enable this, go to System Preferences > Keyboard > "Show Keyboard and Character Viewers in menu bar"