  Semiclassical approximations often involve the use of stationary phase
approximations. This method can be applied when $\hbar$ is small in comparison
to relevant actions or action differences in the corresponding classical
system. In many situations, however, action differences can be arbitrarily
small and then uniform approximations are more appropriate. In the present
paper we examine different uniform approximations for describing the spectra of
integrable systems and systems with mixed phase space. This is done on the
example of two billiard systems, an elliptical billiard and a deformation of
it, an oval billiard. We derive a trace formula for the ellipse which involves
a uniform approximation for the Maslov phases near the separatrix, and a
uniform approximation for tori of periodic orbits close to a bifurcation. We
then examine how the trace formula is modified when the ellipse is deformed
into an oval. This involves uniform approximations for the break-up of tori and
uniform approximations for bifurcations of periodic orbits. Relations between
different uniform approximations are discussed.
