We present the first and so far the only simulations to follow the fine-grained phase-space structure of galaxy haloes formed from generic Λ cold dark matter initial conditions. We integrate the geodesic deviation equation in tandem with the N-body equations of motion, demonstrating that this can produce numerically converged results for the properties of fine-grained phase-space streams and their associated caustics, even in the inner regions of haloes. Our effective resolution for such structures is many orders of magnitude better than achieved by conventional techniques on even the largest simulations. We apply these methods to the six Milky Way mass haloes of the Aquarius Project. At 8 kpc from the halo centre, a typical point intersects about 1014 streams with a very broad range of individual densities; the ∼106 most-massive streams contribute about half of the local dark matter density. As a result, the velocity distribution of dark matter particles should be very smooth with the most-massive fine-grained stream contributing about 0.1 per cent of the total signal. Dark matter particles at this radius have typically passed 200 caustics since the big bang, with a 5–95 per cent range of 50–500. Such caustic counts are a measure of the total amount of dynamical mixing and are very robustly determined by our technique. The peak densities on present-day caustics in the inner halo almost all lie well below the mean local dark matter density. As a result caustics provide a negligible boost (<0.1 per cent) to the predicted local dark matter annihilation rate. The effective boost is larger in the outer halo but never exceeds about 10 per cent. Thus, fine-grained streams and their associated caustics have no effect on the detectability of the dark matter, either directly in Earth-bound laboratories or indirectly through annihilation radiation, with the exception that resonant cavity experiments searching for axions may see the most-massive local fine-grained streams because of their extreme localization in the energy–momentum space.