This letter introduces a framework for evaluation of the losses used in point set registration. In order for a loss to be useful with a local optimizer, such as e.g.& x00A0;Levenberg-Marquardt, or expectation maximization (EM), it must be monotonic with respect to the sought transformation. This motivates us to introduce monotonicity violation probability (MVP) curves, and use these to assess monotonicity empirically for many different local distances, such as point-to-point, point-to-plane, and plane-to-plane. We also introduce a local shape-to-shape distance, based on the Wasserstein distance of the local normal distributions. Evaluation is done on a comprehensive benchmark of terrestrial lidar scans from two publicly available datasets. It demonstrates that matching robustness can be improved significantly, by using kernel versions of local distances together with inverse density based sample weighting.