  Ho\v{r}ava gravity has been proposed as a renormalizable, higher-derivative
gravity without ghost problems, by considering different scaling dimensions for
space and time. In the non-relativistic higher-derivative generalization of
Einstein gravity, the meaning and physical properties of black hole and
membrane space-times are quite different from the conventional ones. Here, we
study the singularity and horizon structures of such geometries in IR-modified
Ho\v{r}ava gravity, where the so-called "detailed balance" condition is softly
broken in IR. We classify all the viable static solutions without naked
singularities and study its close connection to non-singular cosmology
solutions. We find that, in addition to the usual point-like singularity at
$r=0$, there exists a "surface-like" curvature singularity at finite $r=r_S$
which is the cutting edge of the real-valued space-time. The degree of
divergence of such singularities is milder than those of general relativity,
and the Hawking temperature of the horizons diverges when they coincide with
the singularities. As a byproduct we find that, in addition to the usual
"asymptotic limit," a consistent flow of coupling constants, that we called "GR
flow limit," is needed in order to recover general relativity in the IR.
