Propane (aka R290) is quite common in commercial fridges/freezers and now seems to also be seeing use in air conditioning, while Isobutane (aka R600A) is the norm in domestic refridgeration.
Reading around, I'm not convinced that 'a norm' really exists, but it certainly seems that both of those scenarios you describe are quite common. I suspect that one of the reasons why there is not more of a 'norm' is that refrigeration kit tends to last for a good few years, but fashions and practices (and rules/regs) change pretty frequently - so that at any point in time what is in-service tends to reflect the spectrum of changing thinking over preceding years.
CO2 has also been used for air conditioning and commercial refridgeration, it has the distinct advantage of being non-flamable while not being anywhere near as bad for the environment as flourocarbons but it's critical point is a little on the low side. I've seen a claim (which I can't verify) this is the reason for out of service freezers in UK supermarkets recently.
That all sounds reasonable.
Phase change is not strictly necessary for a refrigeration cycle (most aircraft use an air-based cycle to manage cabin temperatures!) but it's very helpful.
Now you're stretching my brain to beyond anywhere I think it's ever been before
As I said in my initial description/explanation of the refrigeration cycle I think I've probably always assumed that the cycle was (at least in practice) always based on the release and absorption of latent heat during phase changes. However, if I understand you correctly, you're now saying that the cycle can work simply on the basis of the temp changes which occur when a gas is compressed or allowed to expand, but is always in the gas phase (I presume that it cannot work if the substance is always in the liquid phase) - and I have to say that, now you've made me think about it, I see no reason why that couldn't be the case, at least qualitatively. However, I somewhat suspect that there could well be issues in relation to 'efficiency' (in whatever sense), since I think that (quite apart from what I go on to say in the next paragraph) latent heat can probably be pretty large in comparison with the relatively modest changes in temp with occur with compression/expansion of a gas?
You may also have helped me a bit to understand a conversation I had a few weeks back. It was with an old friend who is a professor of Theoretical Physics and I had asked him if there is a theoretical upper limit to the efficiency achievable by a heat pump. His answer was along the lines of "there will obviously be a limit if there are no phase changes, since (a calculable amount of) work then has to be done to raise the temp of the gas by compression" But that "if the usual phase changes
did occur, then that would be a much more interesting question that he would have to go away and think about"! I suppose I thought that his first scenario was a hypothetical one which never arose in practice, but maybe I was wrong about that!