  We demonstrate a class of local (Noetherian) unique factorization domains
(UFDs) that are noncatenary at infinitely many places. In particular, if $A$ is
in our class of UFDs, then the prime spectrum of $A$ contains infinitely many
disjoint (except at the maximal ideal) noncatenary subsets. As a consequence of
our result, there are infinitely many height one prime ideals $P$ of $A$ such
that $A/P$ is not catenary. We also construct a countable local UFD $A$
satisfying the property that for every height one prime ideal $P$ of $A$, $A/P$
is not catenary.
