Actually,"earth ground" circuits are the ones you normally look at with Oscilloscopes,where one leg of a power supply is connected to chassis & hence ac ground.("Earth",or "PE")
In that case,both the ""ground" lead on the 'scope & the device ground are at the same potential.
I think you are referring to situations where neither side of the voltage you are probing is at ground potential.
This could be looking at the voltage drop across a resistor in the middle of a network.
There are two ways commonly used to do this:
(1) Look at the voltage w.r.t ground on each end & work out the difference in your head----this is easiest for dc
measurements.
(2) Use the method shown in this video.
(3) Use a differential probe-----expensive but good!
Let's bite the bullet of the big "nasty"---probing the Mains supply.
Note: Only use a X10 or X100 probe for this,& check the specs of both probe & 'scope before proceeding.
Mains supplies usually have three connections---Active,Neutral,& Earth.
In Oz,& various other places,Neutral is connected to Earth at the entry to your house.("meter box")
You can't just call Neutral "near as dammit" Earth,because it carries return current for all the Electrical stuff you have running,& will have a voltage drop due to circuit resistance ,elevating the potential at your GPO above that of the Neutral/Earth junction at the "meter box".
You may just get away with clipping your 'scope probe "earth clip" onto the Neutral line & probing the Active, without any "zaps"BUT
(a) The Electrical Authorities won't like it at all--some of the return current from the appliances on that circuit will flow in the Earth line,which contravenes Regulations.
(b) The RCD will operate,if you have one.
As you are unlikely to be messing around with the fixed wiring,the most likely place you will be looking at Mains is inside a device you have on the bench.
In a perfect world,all devices would be wired & colour coded correctly,so the worst that could happen is (a) or (b) above.
If the device is wired with Active & Neutral reversed,or you misidentify them,you are shorting out the mains supply via your 'scope & its Earth circuit.
Large BANG!--goodbye probe,maybe 'scope,maybe you!
There is a sneaky trick if you just want to check for the presence of Mains in your device----use one X10 or better probe with the earthclip completely removed.
The 'scope is now only grounded via the Earth lead in its power cord & the house wiring.
Some current then flows in the Earth circuit,but it is miniscule.
Obviously this means that any high frequency artifacts won't be accurately displayed,but if all you want to do is see if the Mains are there,it works well.
If you absolutely have to "hang across" Active & Neutral,use the method shown in the video,or preferably,a differential probe,or a portable 'scope of suitable ratings,with insulated connectors.