We present a systematic study of the geometric structure of non-singular spacetimes describing black holes in Lorentz-violating gravity. We start with a review of the definition of trapping horizons, and the associated notions of trapped and marginally trapped surfaces, and then study their significance in frameworks with modified dispersion relations. This leads us to introduce the notion of universally marginally trapped surfaces, as the direct generalization of marginally trapped surfaces for frameworks with infinite signal velocities (Horava-like frameworks), which then allows us to define universal trapping horizons. We find that trapped surfaces cannot be generalized in the same way, and discuss in detail why this does not prevent using universal trapping horizons to define black holes in Horava-like frameworks. We then explore the interplay between the kinematical part of Penrose's singularity theorem, which implies the existence of incomplete null geodesics in the presence of a focusing point, and the existence of multiple different metrics. This allows us to present a complete classification of all possible geometries that neither display incomplete physical trajectories nor curvature singularities. Our main result is that not all classes that exist in frameworks in which all signal velocities are realized in Horava-like frameworks. However, the taxonomy of geodesically complete black holes in Horava-like frameworks includes diverse scenarios such as evaporating regular black holes, regular black holes bouncing into regular white holes, and hidden wormholes.
