Motors are designed for sine wave. On square wave you have many problems and the MSW is a compromised square wave with the peak of a sine and the rms of a sine, achieved by reducing the width to 50% of the square.

For motors the peak flux is higher ( it spends longer at peak). There are lots of harmonics about that affect some motors very badly and hardly affect others. The worst may hit a state where they crawl at the 5th or 7 th harmonic speed and will not get through it. Many motors run hotter but can still be within rating. Some will not sustain full load indefinitely within temperature rating but are ok intermittently.

Some get in a real mess with the starting circuits but if you can start them they may be ok when running.

All series universal motors ( power tools etc will be ok unless there is an electronic speed control).

Motors should be happy on a pure sine inverter as long as there is sufficient starting capability. The average inverter claims a surge rating of perhaps 200% for a few hundred milliseconds. A motor may draw 500% overlaod during starting for several seconds on a large inertia load. Fridges and freezers can be a real pig.

You may need an inverter far bigger than you suspect for some motors, others may present little problem.

In general small pure sine inverters seem able to cope with motor starting better than crappy little engine driven alternators.

I suspect the inverter is more likely to die than the motor if you use large motors on cheap msw inverters.