All the experiments I know, which have been performed to test Bell inequalities, are somehow based on measuring the spin degree-of-freedom of some particle (usually photons, sometimes electrons).
I wonder if there are any tests of non-locality that do not involve measuring some spin variable at all but instead use entanglement in some other degree of freedom (angular momentum, momentum, ...?)?
Usually experimental Bell tests are performed using photon polarization. However, experiments have also been devised to test Bell inequalities using so-called "time-bin" entanglement. Essentially, the quantum information is encoded in the time-of-arrival of the photons. Also, there are Bell-type tests using continuous variables. These continuous variables are really collective properties of large collections of photons. Google searches for "continuous variable Bell test" and "time-bin entanglement" can fill in the details. For another example, see this paper, in which Bell tests are performed using ion traps. In this case, the quantum information is being stored in hyperfine levels. The quantum number classifying these is the total angular momentum from the nuclear spin, electron spin, and electron orbital momentum.
So, the short answer is that Bell tests can be performed with any degrees of freedom that are experimentally convenient for storing and transmitting quantum information. Many different degrees of freedom have been tried in real experiments. These each have their pros and cons in terms of how noisy the resulting data is, how expensive or finicky the experiments are, how unambiguous the experimental interpretation is, etc.
