A single phase motor has a mechanical switch inside that cuts the start winding and capacitor ( if it is a capacitor start motor) out at approx. 80% of full load speed. If you use a VFD on this type of motor at low speeds the start winding and capacitor will be energized and burn up after time. But you can use a VFD on a permanent split capacitor motor as they are designed so both start and run winding are energized all the time and the capacitor gives enough phase shift to start and run. VFD's were designed to vary the speed on 3 phase motors with frequency modulation and constant voltage therefore giving constant torque. You can also buy a single phase VFD that will give you a 3 phase output which comes in handy if you have a 3 phase motor but only single phase power. DC motors can vary speed by a variable resistance in series with the shunt field (called field weakening) if it is a shunt motor. On the other hand a permanent magnet DC motor can vary speed only by varying the voltage with some kind of SCR drive, or a high wattage variable resistor ( not recommended , lots of heat).