Résumé : We study measure-class-preserving (mcp) equivalence relations and seek criteria for their (non)amenability. Such criteria are well established for probability-measure-preserving (pmp) equivalence relations, where tools like cost and $\ell^2$-Betti numbers are available. However, in the mcp setting, where these tools are absent, much less is known. We discuss a recently developed structure theory for mcp equivalence relations, including a precise characterization of amenability for treed mcp equivalence relations in terms of the interplay between the geometry of the trees and the Radon–Nikodym cocycle. This generalizes Adams' dichotomy to the mcp setting and yields anti-treeability results for locally compact groups. We also establish a Day–von Neumann-style result for multi-ended mcp graphs, strengthening the Gaboriau–Ghys theorem. Joint work with Robin Tucker-Drob, and with Ruiyuan Chen, and Grigory Terlov.