You could use a VFD with a capacitor-start induction-run single-phase motor. You'd always have to start at 60Hz and you'd not be able to use the full speed range especially at the low end (and reverse on the VFD would do nothing). You should be able to run any single-phase AC motor from a VFD as long as you never vary the frequency from 60Hz, but why bother?you can't use a vfd with a single phase motor
I see that Invertek markets a VFD for PSC motors.The extent of my knowledge is what I have read & not hands on with VFD & single phase AC motors. I do see them on eBay built for single phase in & out.