  We consider the metric of an axially symmetric rotating black hole. We do not
specify the concrete form of a metric and rely on its behavior near the horizon
only. Typically, it is characterized (in the coordinates that generalize the
Boyer-Lindquist ones) by two integers $p$ and $q$ that enter asymptotic
expansions of the time and radial metric coefficients in the main
approximation. For given $p,$ $q$ we find a general form for which the metric
is regular, and how the expansions of the metric coefficients look like. We
compare two types of requirement: (i) boundedness of curvature invariants, (ii)
boundedness of separate components of the curvature tensor in a free falling
frame. Analysis is done for nonextremal, extremal and ultraextremal horizons
separately.
