If you test a power supply with a live ground such as an offline supply and return , and there is no earth ground in the circuit but you know it is low voltage output, you would use earth ground for two probes and invert channel 2 and Add Ch1&Ch2 to make a differential measurement.
If the supply is transformer isolated and floating you can use probe earth ground on either output if you know the output is < scope Vmax such as 200V.
If you are expecting a live ground fault and have the scope plugged into another outlet without a ground fault then connecting the probe ground may bridge the live ground creating a current fault possibly in the scope earth path to line plug. Thus isolation of grounds might protect the scope ground path but will not protect you, so my advice is use the same outlet as the supply under test. If you are unfamiliar with the site, AND suspect a live ground fault then check device ground with the probe ungrounded.
However understand that if the supply return 0Vdc is floating , it is common to see stray voltages say up to 1/2 of the line voltage if it has a leakage resistance to line equal to the probe resistance to ground. However the current is limited in uA due to resistance. So using the probe earth ground is useful to absorb these low currents and get low noise results.
I recall in mid ‘70’s while working in an Aerospace electronics R&D lab next to a machine shop, I got a healthy shock of maybe 10mA and high voltage connecting a long coax cable from one side of the lab to an instrument on the other side. As it turned out there was a live ground fault for some reason that I never found out. But it did not cause any failure, and it was reported to the Electrician to repair.