  We calculate numerically the periodic orbits of pseudointegrable systems of
low genus numbers $g$ that arise from rectangular systems with one or two
salient corners. From the periodic orbits, we calculate the spectral rigidity
$\Delta_3(L)$ using semiclassical quantum mechanics with $L$ reaching up to
quite large values. We find that the diagonal approximation is applicable when
averaging over a suitable energy interval. Comparing systems of various shapes
we find that our results agree well with $\Delta_3$ calculated directly from
the eigenvalues by spectral statistics. Therefore, additional terms as e.g.
diffraction terms seem to be small in the case of the systems investigated in
this work. By reducing the size of the corners, the spectral statistics of our
pseudointegrable systems approaches the one of an integrable system, whereas
very large differences between integrable and pseudointegrable systems occur,
when the salient corners are large. Both types of behavior can be well
understood by the properties of the periodic orbits in the system.
