we did use a microwave on a modified sine wave inverter OK, but I think it depends on both the inverter and microwave combination in regards to the level of success - I have to say that on this boat I bit the bullet and bought a pure sine wave inverter (Sterling), and although expensive, is a far better bit of kit.
Sure, it will hit the batteries while running, but microwave ovens by their nature, are usually run for short periods, and are the sole 'high power' item I use on the inverter.
Other small wattage items will not cause an issue - especially with a good (high efficiency) inverter.
I use a Mastervolt 2000W / 100A Combi Sine inverter/charger connected to a 3 x 230 Ah battery bank. We run everything on 220V: Chargers, laptop, water cooker, coffee machine, TV, Combi MW (although I've never used the MW setting) even a 1800W Vacuum cleaner.....
OK it did cost quit an amount of € (includes Batt. Monitor and remote control for Max I shore setting), but it's one of the best investments of the ship.