Oscilloscope with isolated ground
The obvious answer is: when you want to do a floating measurement; i.e. not referenced to ground. Now with a common ground oscilloscope, this is a dangerous thing to do as you have to detach the ground within the oscilloscope from the mains ground. This could result in dangerous voltages on the casing of the oscilloscope, and should never be done.
The right way to do is to use a oscilloscope specifically designed to be able to do floating measurements; thus being safe from any high voltages.

...And why do you want to do a floating measurement you ask?
In some fields of engineering, you sometimes want to measure two machines or circuits with respect to each other, but the grounds are not connected. In that case, you don't want to connect the grounds with eachother through your oscilloscope, that would cause large currents to flow.