  We study distributions of the ratios of level spacings of a rectangular and
an Africa-shaped superconducting microwave resonator containing circular
scatterers on a triangular grid, so-called Dirac billiards (DBs). The
high-precision measurements allowed the determination of all 1651 and 1823
eigenfrequencies in the first two bands, respectively. The resonance densities
are similar to that of graphene. They exhibit two sharp peaks at the van Hove
singularities, that separate the band structure into regions with a linear and
a quadratic dispersion relation, respectively. In their vicinity we observe
rapid changes, e.g., in the wavefunction structure. Accordingly, the question
arose, whether there the spectral properties are still determined by the shapes
of the DBs. The commonly used statistical measures, however, are no longer
applicable whereas, as demonstrated in this Letter, the ratio distributions
provide most suitable ones.
