I've got a Microfridge where the fridge plugs into the microwave, and then the microwave plugs into the inverter. When the microwave is activated, the fridge is automatically shut down until about one minute after the microwave is finished. It's designed to never pull more than 10 amps AC due to the wiring in older apartments and college dorms. The microwave is a relatively low power model at 900 Watts, which is 7.5 amps AC (which is 75 amps DC, but it generally pulls closer to 50 amps when running), and the fridge 1.3 amps AC (13 amps DC, but pulls more like 10 amps), and both together is 8.8 amps AC (88 DC), but the microwave will pull more nearly 10 amps at startup, so it kills the fridge automatically.
The only way to know the smallest size inverter you can get away with is add up all the Watts of the appliances you plan to run at the same time, then, take that number and add 10% to itself to account for the 10% amp loss in the inverting process. For example, an appliance that draws 480 Watts, plus 10% (48 Watts) is 528 Watts.
Then, total Watts divided by volts will get you the amp draw. (amps X volts = Watts, and Watts / volts = amps).