Are the fastest moving pieces of a jet engine supersonic?
Yes, the tips of the fan blades.

Let's look at an example: The GE-90 has a fan diameter of 3124&nbsp;mm and a maximum rotational speed of 2475&nbsp;RPM. Their circumferential velocity is d·π·41.25 = 405&nbsp;m/s or Mach&nbsp;1.19 at sea level and ISO atmosphere. Now you need to add the flow speed of the air, but this calculation alone should be sufficient to prove the point.

The inner part of the fan and all the compressor stages belonging to the core engine experience flow at subsonic speed, even if the whole engine is flying at supersonic speed (see picture below). To make ignition as complete as possible, the flow speed is continuously reduced the closer you move to the fuel injectors. Since the air is heated in the combustion chamber, the hotter gas has a higher speed of sound, and all the flow through the turbine is subsonic, too. This includes the low-pressure turbine. Even the 1,600&nbsp;ft/s shown below at the turbine is subsonic, because at 1100°C the speed of sound in air is 720&nbsp;m/s or 2,362&nbsp;ft/s.



Overview of parameters in a typical jet engine.

Only in the rear part of a convergent-divergent nozzle with reheat switched on will supersonic flow again be attained. This is exactly aft of the narrowest section, called the throat, where airspeed is equal to the speed of sound, and the petals of the nozzle are the only moving parts here. And they move rather slowly.