I will discuss quantitative compactness properties of commutators of singular integrals and pointwise multipliers, and their characterisation in terms of the membership of the multiplier function in different function spaces. The quantitative compactness is measured by so-called Schatten norms, which describe the rate of convergence of finite-dimensional approximations of the operator. For commutators, there is a curious cut-off phenomenon: for Schatten class parameters either above, or equal to, or below a dimensional threshold, the Schatten class membership (or its weak-type version at the threshold) of the commutator is characterised by the condition that the multiplier belongs either to a Besov space, or a Sobolev space, or the space of constant functions, respectively. Over the past few years, the original results of this type on the Euclidean space have been extended to many new settings by several authors. In a recent joint work with R. Korte, we obtain a general framework that covers, unifies, and further extends many of these results. Some implications can be proved in general spaces of homogenous type, while others require additional standard assumptions, like suitable Poincare inequalities. As a tool of independent interest, we also obtain new characterisations of Sobolev spaces in this setting.