Which priority queue implementations are stable?
Here's how I would suggest that you approach the problem.  Try running each data structure by hand on some small examples.  Then, based on those examples, see if you can form any conjecture about which data structures are stable and which ones aren't. For the ones you suspect are stable, try proving it.  For the ones that you suspect are not stable, try to come up with a counterexample to demonstrate that they're not stable.