And there's an emerging trend of "natural" refrigerants like hydrocarbons, water, and carbon dioxide. The latter two require special parts and would not be suitable for DIY use, but hydrocarbons are very good for DIY use.
Most common is R290 (refrigeration propane or dimethylmethane) and mixtures based on it such as R433b. (Note that it is much higher purity than camping propane, though there have been some attempts at purifying camping propane into refrigeration propane.) Main disadvantage is the flammability, but as the oil makes all refrigerants flammable, it's actually a much lesser problem than you might think for the small systems we typically work on. The interesting part is that hydrocarbons are actually very popular overseas.
R433b (often sold as ES22a) is designed as a general replacement for R22 in air conditioning, but it does work for heat pumps as well. R433a and R433c (often sold as ES502a) is somewhat better tuned for heat pump and refrigeration applications.