There are a few different options.
Applying high voltages to the scope directly is not recommended. Sure, most scopes are perfectly capable of handling a few hundred volts, on recent ones it's usually printed next to the inputs together with a CAT number. Usually the number will be CAT I, only for use on devices isolated from the mains, but CAT II and up are sometimes also seen. In principle, this means you can connect it to mains power, as long as you keep in mind that the outer BNC conductor is connected directly to ground on most scopes, and all outer conductors are thus also connected together (there are exceptions, but it's true for the vast majority of scopes). Making a connection between the outer shield and a live wire will at the very least trip a circuit breaker somewhere. If you're unlucky, it will also destroy part of the equipment.
Some scopes have 50 ohm, or switchable 1 Mohm / 50 ohm inputs. The 50 ohm ones can usually only handle a few volts at most (average over time) before the internal resistor overheats.
The next step up are passive probes, the usual stuff that comes with many oscilloscopes as a standard accessory. Often 10x, sometimes switchable 1x/10x. They're also available in 100x, 1000x, but those aren't usually delivered with the scope. They're also single-ended, the reference wire connects to the outer conductor (ground) so all the same warnings as for connecting something directly to the scope apply.
What can be done if you need a differential measurement is measuring the difference between two channels, each with the ground lead clipped to ground. Some folks simply clip the ground leads of both probes together, leaving you with two crocodile clips dangling together between your probes, just begging to short something out by accident. If a ground point is available at the device being tested, you might prefer to use that instead. If one isn't available, perhaps see if you can get good results without ground leads at all (will pick up noise). Taking the difference between two channels obviously requires that both channels have the same voltage setting. Accurate measurements also require both probes to be perfectly matched, and both oscilloscope channels to have identical performance and good common mode rejection (CMRR). They probably don't, so this is only useful for measurements that do not require high accuracy or with little common mode signal.
1x passive probes are useless if you want to measure voltages higher than what the scope can handle on its own, as they are just direct connections without attenuation.
10x passive probes may offer a somewhat higher voltage range than the scope itself, but usually not by much. It's rare to see one rated for more than 600V. 100x probes often do allow significantly higher voltages, I've got one that is specified for up to 2.5 kV for example. These are quite similar in physical appearance to 10x probes, though you might notice somewhat more insulation enclosing the terminals than on a 10x probe. Price is reasonable too, perhaps two or three times what you'd pay for a good 10x. Perhaps worth having even if you don't habitually measure high voltages because these also offer a higher impedance at the probe tip, and usually have fairly decent bandwidth too. 1000x probes tend to be bulky and expensive monsters with poor bandwidth. For 10kV and higher voltages usually. Only get these if you really need them.
Probes may also have a CAT rating on them, just like the scope. If you have a passive CAT III probe, and the scope is CAT I, stick to the lower rating, as it's really a direct connection.
Also, be very careful when using a passive probe at voltages higher than those the scope itself is rated for. For example, suppose you are using a 100x 2.5kV passive probe to measure 2kV on a scope rated for 400V. That's fine as long as you keep the oscilloscope DC coupled. If you accidentally switch to AC coupling, you can destroy the AC coupling capacitor in the oscilloscope, and perhaps a bit more of the input circuitry too depending on what happens to be near that cap.
Then, the better option (well, certainly safer). Active probes, which have active components built in and need a power supply or batteries. There are some single-ended and differential ones for high speed low voltage work. Not useful here. The ones you want are differential high voltage probes. These are high voltage amplifiers that take the difference between two inputs, attenuate it, and convert it to a single-ended output, usually a BNC cable that connects directly to the oscilloscope, or through a 50 ohm terminator (depends on the probe). The inputs and the output are (mostly) isolated from each other (some very high impedance connection may exist, but it will be almost negligible). That also means that the CAT rating of the probe applies when making measurements.
Lots of variations available with different maximum bandwidths, measurable and maximum differential and common mode voltages, attenuation factors, input impedance, CMRR... Better specifications will cost you more, and they're not exactly cheap to begin with. Be sure to check their specifications thoroughly before buying to see if they meet your needs.
Oh, and also consider derating, for both active and passive probes. The maximum permissible voltage is for DC and low frequency measurements. Most probes will not meet that specification at high frequencies, and if your probe comes with a manual it will probably contain a table or graph showing how high the maximum allowable voltage is at various frequencies. For differential probes, there will also be a derating curve for CMRR.
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