  In this work we study the local behavior of geodesics in the neighborhood of
a curvature singularity contained in stationary and axially symmetric
space-times. Apart from these properties, the metrics we shall focus on will
also be required to admit a quadratic first integral for their geodesics. In
particular, we search for the conditions on the geometry of the space-time for
which null and time-like geodesics can reach the singularity. These conditions
are determined by the equations of motion of a freely-falling particle. We also
analyze the possible existence of geodesics that do not become incomplete when
encountering the singularity in their path. The results are stated as criteria
that depend on the inverse metric tensor along with conserved quantities such
as energy and angular momentum. As an example, the derived criteria are applied
to the Plebanski-Demianski class of space-times. Lastly, we propose a line
element that describes a wormhole whose curvature singularities are, according
to our results, inaccessible to causal geodesics.
