There are VFD boxes made for speed control of single phase motors by reputable companies. One from Invertek here **LINK** .
However these are generally only suitable for permanent split capacitor and capacitor start and run motors which are effectively two phase machines. As such they are effectively self starting and, given suitable manipulation of the relative phase and currents in the two windings able to generate useful power over a reasonable speed range. Power drop off away from rated speed will be more severe than with a three phase motor and useable speed variation considerably less. Optimal set up is probably a nightmare because the two windings have different impedances and carry different currents.
If a single phase motor is already running and the starter switch disabled it is possible to get a limited speed variation by changing the input frequency. Whether by VFD or other means. In practice the useful speed change range is only down to where the motor would normally stall out or re-engage the centrifugal starter winding switch. Even then efficiency will be poor with power dropping off rapidly. A similar speed increase range may be possible but again with serious power and efficiency implications. In general single phase motor design compromises are aimed at keeping run speed close to the nameplate rating and holding torque up to combat load changes without serious speed change. Outside the design range horrible things tend to happen. Exactly what depends on detail motor design. There are a lot of ways of doing single phase motors.
The inexpensive single phase motor driving VFD boxes seen on E-Bay et al appear to be simple input frequency variation devices with some means of temporarily driving the start windings to get the motor running. As such, assuming the starting device works, they may be capable of reliably running the common 50 Hz single phase motor between 40 and 60 Hz. Useful power range being most likely less. If you get lucky with a motor and drive pair that happen to suit each other things might work out acceptably well for unambitious applications but I don't see the risk as worthwhile.
Far better to bite the bullet, get 3 phase motor and VFD from a proper, reputable, supplier with back up services just in case things go wrong and know you have something that just works. Being frugal is admirable, and usually necessary for folks like us, but there are times when you have to accept that things cost what they do for good reason. Often "too cheap" = "too cheaty".
Clive