  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
(Ho\v{r}ava-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 Ho\v{r}ava-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 Ho\v{r}ava-like frameworks. However, the taxonomy of
geodesically complete black holes in Ho\v rava-like frameworks includes diverse
scenarios such as evaporating regular black holes, regular black holes bouncing
into regular white holes, and hidden wormholes.
