Why are there no supersonic propellor driven aircraft?
The NASA has done research on the supersonic propeller at the end of WWII, which can be found here and here.

The main reasons that we don't see any supersonic propellers are the following:


The supersonic regions at the blades create tremendous forces which will tear the propeller apart. Thus the design needs to be such that these forces can be mitigated. This is also the case for jet engines, which is why the ducts are designed such that the airflow into the engine is slowed down to subsonic speeds.
The developments in jet engines were going at such a fast rate that the propeller was soon surpassed and was written off. However, since turboprops are much more efficient than jet engines they might pop up on the research radar in the near future.