I use an AMES sine inverter to power a laptop charger, but it won't run clocks because the frequency is crystal controlled and stuck at 59.1 Hz
Just because it's sine wave doesn't mean it's equal to the mains, but square wave inverters are junk as far as I'm concerned. I had a couple and they wouldn't run a 25 watt tabletop fan properly. It hummed loudly and ran off-speed.
Running a microwave from an inverter is borderline abuse. Microwave ovens have a huge coil that a square wave will not work with properly. Microwave ovens also use HIGH voltage to get the tube going and square wave inverters don't have the same peak voltage as the mains. The area under the curve may be the same, but for a step up transformer that's not enough.
For all the money you've spent on inverters, you could have bought a 12v microwave.
http://www.outdoorgb.com/c/12v_kettles_cookers/