  We discuss different kinds of Killing horizons possible in static,
spherically symmetric configurations and recently classified as "usual",
"naked" and "truly naked" ones depending on the near-horizon behavior of
transverse tidal forces acting on an extended body. We obtain necessary
conditions for the metric to be extensible beyond a horizon in terms of an
arbitrary radial coordinate and show that all truly naked horizons, as well as
many of those previously characterized as naked and even usual ones, do not
admit an extension and therefore must be considered as singularities. Some
examples are given, showing which kinds of matter are able to create specific
space-times with different kinds of horizons, including truly naked ones. Among
them are fluids with negative pressure and scalar fields with a particular
behavior of the potential. We also discuss horizons and singularities in
Kantowski-Sachs spherically symmetric cosmologies and present horizon
regularity conditions in terms of an arbitrary time coordinate and proper
(synchronous) time. It turns out that horizons of orders 2 and higher occur in
infinite proper times in the past or future, but one-way communication with
regions beyond such horizons is still possible.
