While you’re probably fine using an inexpensively modified sine wave inverter in your camper, there are a few different things that you don’t typically want to run off a modified sine wave. First of all, anything that uses an AC motor isn’t going to work to full capacity on a modified sine wave. Appliances like refrigerators, microwaves, and compressors that use AC motors won’t run as efficiently on a modified sine wave as they would on a pure sine wave.
In some cases, running an AC motor on a modified sine wave may actually lead to a buildup of excess waste heat that could damage the equipment.
You’re probably okay to use these devices with a modified sine wave inverter, but you do so at your own risk.
The other main thing to be concerned about with modified sine wave inverters is delicate medical equipment. For instance, if you use a CPAP to help correct apneas when you’re asleep, you’re going to be better off with a pure sine wave inverter. Some CPAP manufacturers warn that you can damage your machine with a modified sine wave inverter, and others specify that the CPAP will work but the humidifier unit could be damaged.
Other types of medical equipment, like oxygen concentrators, also need a pure sine wave.
In these types of cases, you’re better off using pure sine wave power or looking for a unit that can be powered by DC without the need for an inverter at all if it’s available.
Of course, some devices can suffer from unwanted interference from a modified sine wave inverter. You mentioned powering your radio off your modified sine wave inverter, and that’s one case where there might be an issue. While it’s unlikely that you’ll damage your radio, it may pick up interference from the modified sine wave, which could make it real hard (or even painful, depending on the degree of interference) to listen to the ballgame after you’re done fishing for the day.