  Properties of a quantum mushroom billiard in the form of a superconducting
microwave resonator have been investigated. They reveal unexpected nonuniversal
features such as, e.g., a supershell effect in the level density and a dip in
the nearest-neighbor spacing distribution. Theoretical predictions for the
quantum properties of mixed systems rely on the sharp separability of phase
space - an unusual property met by mushroom billiards. We however find
deviations which are ascribed to the presence of dynamic tunneling.
