Cheese – Parmesan rind: should I use it, or trash it

By the wording of the question, I'm pretty sure you're using one of those pyramidal graters or a "sheet" grater.

That's great for getting the "bulk", but if you want to grate a small amount of cheese (or the last bit of a large amount, if you don't want to just eat it) then you really need to get yourself a rotary grater. They're more commonly used for fine/hard cheeses such as Parmesan, but most half-decent ones will come with a "coarse" blade that can be used for cheddar, mozzarella and so on.

Rotary graters all have a plate at the top that lets you push the cheese down onto the blade. Higher-quality graters will give you slightly better results; the $10 cheap ones tend to leave a mush at the bottom that's hard to grate, even if you squeeze really hard, but that's easily remedied with a small wooden block or really any object at all to put between the top plate and the cheese you're trying to grate.

I usually freeze mozzarella and then grate it (longer the better, unless you are in a hurry, then 20 minutes or so works OK). This works very well. Other soft cheeses, such as those meant to be eaten at room temp, brie, for example, I wouldn't freeze... Of course, I don't think many of us are grating brie anyway.