For most propellors, when the tip speed gets much above Mach 0.8 or 0.9, the efficiency drops off pretty rapidly. Also, the noise level becomes very high. Larger diameter and slower turning propellors are more efficient for takeoff and climb performance, anyway.

There are a few designs that used supersonic propellors (as opposed to the plane itself). For example, back in the 1950s Republic tested the XF-84H that was powered by a big Allison turboprop engine. The propellors were designed for supersonic speed. The idea flew but it really wasn't worth the bother. It was said to be perhaps the noisiest plane ever built. The normal name for an F-84 was the Thunderstreak. The XH-84H was called the Thundershreak. There were also some supersonic propellor tests on the XF-88B.

I've heard that the propellor tips on the Tu-95 "Bear" bomber can exceed the speed of sound at high cruising speeds. It's incredibly noisy and they seldom do that, or at least that's what I've heard. Unofficially, the Bear was the fastest propellor driven plane ever made with a max speed at 25,000 feet of 575 MPH. There was an airliner derivative of the Bear called the Tu-114 that used the same engines.