The rings are only a fraction of the rocks in a solar system. There's hundreds of thousands of them careening all over the place. If you look up "Amor Asteroid", you'll see a belt of them that mostly traverse the orbit of Mars. Deimos & Phobos are likely amor asteroids that got caught by Mars. Check out the 2009 Jupiter impact event:
http://en.wikipedia.org/wiki/2009_Jupiter_impact_event
This ought to answer your question about the stability of rings.
Generally, though, measured over human lifetimes, space rocks are very stable. Judged by planetary lifetimes, well, look at the moon and decide why it is cratered so. Likely something sooner or later will come into impact with the rings. Rings don't just magically appear: they are themselves the result of a planetary impact. So if one thing can come along to make rings, certainly another could travel the same path.
After all that, you could also measure stability in stellar lifetimes instead of planetary ones. Stars inhale and exhale sort of like anything else that lives and breathes. If they cannot breathe anymore, they fall apart and die, a truly simplistic explination for a complex phenomenon. Main sequence suns like Sol will expand and eat a few planets before they cool off.
Believe it or not, there exist plans to save the Earth from stellar expansion, which should occur in roughly 4-5 billion years. The big minds are considering strapping engines to the Moon and using it as a gradual slingshot to break Earth out of solar orbit. Several thousand passes ought to be gentle enough to break the Earth free without destroying everything. But there are also plans to mount engines on the continents and go at it that way. Tough luck, Australia.
We have to act fast, though. Just because the Earth may be swallowed by the Sun in 4 billion years doesn't mean it won't get sweltering hot, first. Earth will probably endure Venus or Mercury like surface temperatures in under a billion years anyways. Even Canada. Scotland might be nice, though: no more crazy horizontal rain all the time.
EDIT: Thinking more on the stability of rings, it's the people that are the problem. Consider that in ED there is more than enough space for everyone, yet everyone is constantly at war. So you set up Damocles City on ringed planet Twelvefield, sooner or later some military genius will hit upon the idea of strapping cheap engines onto the rocks and bombarding Damocles with stones from space. Each rock would hit with many hundreds of times the force of a nuclear blast. Maybe in the far future there's a defense for that, but even that far ahead, Sir Isaac Newton will still be relevant.