I heard a "pop", and my Patton U2-1487 floor fan stopped. How can I test the switch to see if it needs replacing?

Just guessing, but I doubt it is a switch.

More than likely it was the motor speed capacitor.

If the fan is inexpensive, just replace it.

Make a jumper and bypass the switch. If the fan works, it's the switch. If not, it's the motor.