Actually was going to ask about mains, can you actually stick it in mains or do you need to do anything special? Obviously you'd want to either not use the ground clip, or ensure that the ground source you use is the same as the scope's ground. Or isolate the scope and plug into a GFCI, but probably best not to isolate it. Would be neat to test out UPSes and such to see what kind of output they produce. Some UPSes do funky things with the grounding so would just want to watch with that.
You really shouldn't. The oscilloscope is rated CAT I, which generally speaking means its input cannot be directly connected to the mains, as the input is not rated to handle over-voltage transients typically expected on such a circuit.
You can use a CAT II minimum rated 10X probe (600V or higher), or an isolating step-down transformer with sufficient series resistance to measure the mains with an oscilloscope. (Not that it's much use - it's just a sine wave with some rather flat-tops caused by all the low PF simple full-wave rectified supplies...)
You -should- be able to probe an isolated UPS output with a 10X probe, as long as it is operating on its internal batteries only.
In general, you don't want to go about floating your scope. The reason being you lose a large part of the protection that a grounded scope offers you. If you float your scope, everything ground on it can be potentially live. With a good scope, if you happen to connect the ground to a live high-ish energy source, the worst damage you'll usually do is blow up a probe clip and/or trip a breaker. Maybe the scope could be damaged, but this is rare because the shielding of most oscilloscopes, and the BNC connector, offers a low impedance to ground, bypassing sensitive input circuits. But, if you remove this grounding, you can touch any metal part of the scope and it could be live, without you knowing.
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