Many of the older things with transformers in them to produce the power supply work well enough on MSW. Modern things often use series capacitors instead of a transformer and the reactance to a square wave will lead to an excessive current and it may mess up the power supply.
Some of the newer power factor correction rectifiers in better products will not cope with the MSW at all.
It is always a gamble what happens on MSW.
It is really optimistic to call these things modified sine, I prefer to call them modified square wave. They are cheap and cheerful and work for many appliances but there are things that don't work at all or work badly and some things just go up in smoke.
Things are designed for grid operation and the voltage, frequency and waveform are all well defined, it is just luck if you choose something that will tolerate a waveform far removed from what it is designed for.
As electronic products are now being designed to avoid transformers you will probably find more of them blow up on msw. It may be a useful idea if various people listed the things that definitely don't work on msw but even then the things may be specific to certain regions as there may be different power factor and rfi restrictions on products in different part of the world.
As electronics advances it may be that the sine wave inverter will soon come down in cost and we can forget these square wave things all together.
Flux