The currently selected answer is quite contentious with good reason. It's implying that Modified Sine Wave (MSW) will just work without describing why it may or may not work. We can attack this problem from a theoretical/mathematical perspective assuming a load "desires" a pure sine wave.
Assuming a load requires a pure sine wave, that would mean that it rejects or can't use all frequency components outside of the main frequency (50hz or 60hz). If we assume the load is purely resistive (to approximate power wasted) and we normalize it, we can integrate a squared sine wave vs a squared square wave vs a squared MSW to compare power using \$P=V^2/R\$.
Two level square wave (-vdd,+vdd)
$$ Ideal =\int_{0}^{\pi}(sin(x))^2 dx= \pi/2$$
$$ Square = \int_{0}^{\pi}1 dx= \pi $$
$$ Wasted\_power = 50\% $$
MSW, transition happens at \$0.5vdd\$ crossing, aka \$30\$ degrees or \$1/6\pi\$ and \$5/6\pi\$ (-vdd, 0, +vdd):
$$ Ideal=\int_{\pi/6}^{5/6\pi}(sin(x))^2 dx= 1.480 $$
$$ MSW = \int_{\pi/6}^{5/6\pi}1 dx= 2/3\pi = 2.094 $$
$$Wasted\_power = 29.3\% $$
The main concern here, is that if your system truly desires a sine wave input, you're burning a lot of extra power which has the potential to thermally destroy or pre-maturely age your device/system. If you know your system is designed to handle the dumping of 50% more power if it's a square wave or 30% more power in a 3-step MSW then you're probably safe to use a MSW inverter.
Note that this will also cause torque ripple in A/C motors due to off times followed by a hard application of power which increases noise and vibration on top of the thermal issue.
This is mostly an analysis of inductive loads including motors/compressors/fans/transformers(microwave). Note that in the transformer/microwave case, this will cause the transformer to warm up rather than dumping the extra power into your food.
The other category that the OP discussed are digital electronics requiring switch mode power supplies. They should all be using isolated power converters. If we take the flyback converter as an example, it is already switching the primary side coil creating a hard edge into the transformer. More hard spikes aren't going to significantly change the power conversion there besides confuse the feedback circuitry some. The main concern here then is the passing of odd harmonics of a square wave through the flyback transformer which then aren't properly filtered. This could cause more noise on the output, but since the mains frequency is much lower than the converter switching frequency, it shouldn't cause significant power spikes down the line. It would more than likely cause voltage drooping so if you have an application that would be sensitive to lower frequency electrical noise (audio equipment), I would avoid a MSW inverter.
Last note: An MSW may not raise the power to Vdd. Instead it may raise it to \$ 1/\sqrt{2}~=0.707 Vdd \$ the RMS or some other value to improve efficiency or to better match the expected sine wave. It may also reduce the on-time of the MSW as well which would increase efficiency but reduce total extracted power. My calculations above are somewhat worst-case scenarios.
