Recent work has focused on nonparametric estimation of conditional treatment effects, but inference has remained relatively unexplored. We propose a class of nonparametric tests for both quantitative and qualitative treatment effect heterogeneity. The tests can incorporate a variety of structured assumptions on the conditional average treatment effect, allows for both continuous and discrete covariates and does not require sample splitting. Furthermore, we show how the tests are tailored to detect alternatives where the population impact of adopting a personalised decision rule differs from using a rule that discards covariates. The proposal is thus relevant for guiding treatment policies. The utility of the proposal is borne out in simulation studies and a re-analysis of an AIDS clinical trial. This is joint work with Mats Stensrud, Riccardo Brioschi and Aaron Hudson.