http://gassend.net/spaceelevator/breaks/
In summary, if the cable breaks at the base, the elevator will fly harmlessly into space. If it breaks at the counterweight, a large portion of the cable (several thousand miles long) will collide with the Earth’s equator.
The space elevator would become a space whip. I wonder whether this has a use case, like getting hardened unmanned space probes to speeds required for interstellar missions, or just much faster missions to other planets in the solar system.
Assuming a catastrophic break near the upper end of the cable (before the counterweight), there's not much you can do to keep it from coming down.
And just a cable falling could be a significant disaster by itself.
The travel pods, likewise, can always have rockets on them whose sole purpose is to get the passengers to safely. Yes, it means you’re taking up more mass than you strictly need to, but the great thing about a space elevator is that the mass is no longer a hyper-critical parameter.
But yes, no rails here.