I don't know how many different things should not be run on the Modified Sine Wave inverters, but all the low end (MSW) Uninterruptible Power Supplies (like APC Back-UPS model line) state that they should only be used to power PC's - which have switching power supplies in them, and apparently don't mind Modified Sine Wave power. For other loads, APC recommends their sine wave models - like Smart-UPS, or better.
*nod* The SmartUPS units are what you want. Fortunately there was this dot.com boom a few years back and a lot of SmartUPS units are now being sold used at hamfests. I have a total of three 2200's, a 1400, a 700, and a Matrix 3000 (which has a real transformer and can power anything at 220 volts). All of the SmartUps devices cost less than $20 each, all worked fine with new batts, etc.
I would think it is similar for power inverters - and maybe it all depends on how many steps in the wave - how closely it approximates a real sine wave.
The ultimate sine wave inverter is a rotary one.
I think the 12vdc powered MSW inverters can power drills and similar if they are universal motors - don't know if extra motor heating or other side effects occur.
No problem there. I use a BackUPS 600 in my shed to power small stuff, never had a problem. Lately I have been dragging around a little backUPS 400 in the front of the E20, wired to the front batteries (pull them down a bit) for things like weedwhackers. Works fine.
The real winner is the Matrix 3000. It does weigh a ton, but in the bottom of it is a real authentic transformer that can handle just about any start up surge or load.
Chris