My power inverter produces trapezoidal (TPZi) waveforms instead of sine waves. Is it safe to run my PC on this inverter?
There should be no issue - the only part of the computer that might care is the power supply, and that should be fine. From the part-way through the PSU it's all DC anyway, and stabilized and such.
Using an inverter is highly inefficient anyway, and I would recommend replacing the power supply with a 12v (or 24v depending on your circumstances and intended use) power supply if you intend to use the computer on low voltage alot.
Depending on how powerful your computer is you may get away with scrapping the PSU altogether and just having a little adapter plug for the motherboard that converts 12v into the ATX set of powers. There are more powerful arrangements for more meaty systems, most of which are aimed at running a PC in a car, so handle such things as shutting down the computer on loss of accessory power (ignition turned off) etc.