When is it necessary to connect all scope probes to GND (instead of just one)?
I would have thought the answer is obvious: When you need accurate low noise results.

Half the time you just need a quick look to see if a signal is there or not, there is normally no need for a ground connection in that situation.
But if you actually want to measure anything you need a ground connection. The more sensitive the measurement (both in terms of amplitude and frequency) the better the ground needs to be. In this situation closer to the point of interest and shorter wires/connections is generally better.

And for future reference, those jaws are called a Crocodile clips in English.