What BobNH said is the primary problem with existing HVAC residential compressors, they are positive displacement and therefor not suitable for variable speed.
There is also a problem which what is called "slugging" when you start messing with refrigerant compressor speed, regardless of the type of compressor. If the refrigerant cannot absorb enough heat from the coils, the fluid fails to vaporize and gets back to the compressor as a liquid. Since liquids can't compress, the remaining small amounts of vapor get severely over compressed in the compressor and the extreme pressures start to exceed the capabilities of the seals, valve etc., destroying the compressor. Anyone considering using variable speed on a refrigeration compressor had better know at LOT about this issue before attempting it. Experiment failures are extremely expensive.
For everyone's edification, there are VFDs designed to work with single phase motors, but they only can work with 2 specific types, PSC (Permanent Split Capacitor) and Shaded Pole. Unfortunately, the vast majority of installed residential HVAC compressors are Cap-Start / Cap-Run type single phase motors, one of the types that you cannot use a VFD on.
New residential HVAC systems designed to meet the upcoming efficiency requirements are being designed to use small scroll compressors, the most efficient design that works like a centrifugal pump and therefore more amenable to having variable speed applied. Some of them are being built with small 3 phase motors and VFDs that convert the residential 1 phase power to 3 phase just for the motor. That will likely be the norm some time in the future, but that doesn't help with anything already installed.