I know that breaking it on the hot (copper) side allows the one receptacle to be controlled via switch while keeping the other always hot. Are there scenarios where you would ONLY cut the neutral side tab and leave the other? What about scenarios where both would be cut?
Breaking the tab on the neutral side does exactly the same as the hot side: it separates one terminal and neutral outlet from the other terminal and outlet.
I can't imagine a scenario where only the neutral would be cut.
The neutral should be cut where the corresponding hots are cut and come with their own neutral. For example, if the two halves of the outlet are controlled by two separate switches and the power comes from the switches, then the neutrals should be separate. The neutral should not be cut where the power comes in to the box with the outlet, and the wire from the box to the switch is switching the hot and returning it over the white wire.
As the other answer stated already, you would never break only the neutral side tab. You could, but it wouldn't do anything useful.
An occasion when you might have both sides broken, in addition to the aforementioned reason, is if one side is using a different set of neutral circuits than the other. For instance, in a laundry room, I have encountered one side of a receptacle powering the dryer and the other powering the washer. The dryer had a black hot and the washer red; their respective neutrals were connected to different breakers. This prevents a single circuit from being overloaded when both are in use. Connecting them to each other, however, as in a typical receptacle, results in the immediate tripping of the breaker.
In short, don't worry about breaking the tab on a neutral side unless you're an experienced electrician, or you found it that way (perhaps while replacing the receptacle).
Don't give up, skeleton!