Russell's answer is excellent as usual, I just want to add a bit extra.
Most oscilloscopes have a "standard" at least as far as impedance goes (1 Megaohm - note some have a 50 ohm input but that is less common and not relevant here)
The amount of protection and rating of front end components can vary pretty widely from what I've seen. For instance I have seen schematics for scopes rated for something like <50V input with no protection other than a 10k resistor in series with the opamp input.
In comparison you can get old (and probably new although I haven't seen inside one) tektronix scopes with >600V rating and heavy duty protection.
The only safe way to know what the limit for your scope is to read the manual carefully. If they sat you can probe mains voltage with the probe set to 1x then it should be fine - if it's under warranty and it breaks then you are covered anyway. However, heed Russell's advice about transients - if you have to probe mains voltages, whatever the input is rated for I would use a probe with 10x or 100x setting only, so you can't accidentally set it to 1x (see below)
Personally, I rarely probe anything high voltage on my DSO (OWON SD8202) - I use my big old tank of a scope (Tektronix 7633) for stuff >100VAC with a 10x probe and the DUT run from an isolation transfromer. I must admit a long time ago I accidentally used 1x probe setting for 230V (UK) mains on the Tek a few times and it never complained, although I certainly wouldn't recommend this to anyone - I mention it just to give an idea of how well these things were built (guess they assumed some idiot was gonna come along and do silly things like this :-P )
As far as the ground clip goes, on most plug in (to wall plug) scopes this is directly connected to earth ground.
In floating (i.e. no connection to mains ground through anything - USB, charging leads, etc) battery powered scopes with plastic cases then floating measurements may be taken, but as always, follow the manufacturers advice.
What this means is that if you attach the ground clip to anything that is ground referenced (like the mains live wire) and at a potential higher than earth ground, it will create a low impedance path for current to flow (i.e. a short)
Ground referenced means, that one side of the potential is connected to ground - with mains voltage, when the utility wires come into your house, they are split into live, and neutral/earth (which are both connected to each other)
The earth wire is at the same potential as the neutral, but is not meant to carry current under normal circumstances - if there is current flowing in it (for instance if a live wire has fallen against a metal chassis connected to earth) then there is a fault.
If you isolate the ground referenced mains voltage using a transformer, then (as long as the secondary has not been connected to earth) you can connect your ground clip to either side of the secondary and be safe, as the current does not "want" to flow through it (aside from a small amount of capacitive leakage current)
If in any doubt, a good idea is to measure to see if there is any common reference between your ground clip and whatever it is you want to connect it to.
For example, say you have an unknown power supply with two leads and you want to figure if they are ground referenced - one way is to connect one multimeter probe to ground clip and the other to either wire to see if you get any voltage.
Another way is simply to unplug the unknown supply and measure continuity from it's earth plug pin to the output connections - if there is no continuity (or extremely high, say > 1 Megaohm) then there is no reference to ground.
Just in case it's a transformerless supply (or just a badly designed one) you should check that there is no continuity from the live and neutral pins too.
If still in any doubt, don't connect anything up until you completely understand everything.
There are also differential probes (example) you can buy for any scope that can be used to measure the difference between two floating voltages.
Here are a couple of references on grounds/probes:
Tek reference on probe grounds
All About Circuits worksheet on scopes read all of this and the answers to the questions (press reveal)
All About Circuits - Electrical Safety - not about scopes, but very useful information about electrical safety. The section on "Safe Circuit Design" is particularly relevant. Note this does not deal with isolation transformers though (although there is plenty on transformers on another part of the site)
One simple example
Let's say you have two scope probes. Probe A is correctly grounded to the circuit you are measuring but probe B doesn't have a ground connection. Let's also say that probe A is measuring a 1Vp-p squarewave on the circuit and probe B is looking at a DC level.
Despite scopes having a 1M\$\Omega\$ input impedance, there will be currents that flow through the probe due to the edges of the square wave - the scope input impedance is also likely to comprise 20pF capacitance (smaller for x10 probes usually).
This capacitance "takes" a pulse of current down the probe lead every time the square wave alternates causing a small volt-drop down the lead. This won't matter for probe A but, this will appear as a small "spike" on channel B's display because the glitch "becomes" in series with the probe B signal due to it being ungrounded.
If probe B were correctly grounded at the circuit this "glitch" won't be present on the scope B waveform.
What point/frequency does this become significant? I would answer that by saying just don't take short cuts. If you are probing a circuit for whatever reason, use best practise and ground both probe ends at the circuit you are measuring or expect anomalies.
If probe A is looking at 1Vp-p squarewave and probe B is looking at a 5mV analogue signal you'll have troubles on channel B.
Best Answer
Grounding loops are not your problem. The real problem is the unknown potential between the scope and the DUT. Without an Earth GND yes both are floating. (Does your scope even accept a 2 prong input without an Earth GND? Usually, most scopes have the chassis connected to Earth GND.)
If you just probe your DUT with the scope floating, you will attach the scope GND to the DUT GND, making that the common ground. The problem is that if the two are at a different potential then you are not protected from shock and may damage the equipment.
Read this document by Tektronix.
"The four safe solutions for these issues fall into four categories: battery-powered oscilloscopes, differential measurement systems, isolated-input oscilloscopes, and monolithic isolation amplifiers." source
Alternatively, if you can guarantee the Vrms of your DUT signal you could do a differential measurement using 2 single ended probes by connecting their GNDs together and putting one on DUT GND and 1 on the signal. Then doing a math function on the Oscope to subtract the two.