We present a new and completely general technique for calculating the fine-grained phase-space structure of dark matter (DM) throughout the Galactic halo. Our goal is to understand this structure on the scales relevant for direct and indirect detection experiments. Our method is based on evaluating the geodesic deviation equation along the trajectories of individual DM particles. It requires no assumptions about the symmetry or stationarity of the halo formation process. In this paper we study general static potentials which exhibit more complex behaviour than the separable potentials studied previously. For ellipsoidal logarithmic potentials with a core, phase mixing is sensitive to the resonance structure, as indicated by the number of independent orbital frequencies. Regions of chaotic mixing can be identified by the very rapid decrease in the real-space density of the associated DM streams. We also study the evolution of stream-density in ellipsoidal NFW haloes with radially varying isopotential shape, showing that if such a model is applied to the Galactic halo, at least 105 streams are expected near the Sun. The most novel aspect of our approach is that general non-static systems can be studied through implementation in a cosmological N-body code. Such an implementation allows a robust and accurate evaluation of the enhancements in annihilation radiation due to fine-scale structure such as caustics. We embed the scheme in the current state-of-the-art code gadget-3 and present tests which demonstrate that N-body discreteness effects can be kept under control in realistic configurations.