There are a number of real systems that have been extensively studied in the lab that show either classical/ray chaos and quantum/wave chaos. Typically the measurement involves analyzing the higher order modes or eigenfunctions of the system (in the quantum/wave case).
These examples include the studies of Wada boundaries in ray scattering in a system of 4 reflecting balls (by James Yorke's group--made the cover of Nature, below), the measurement of scars in the bunimovich stadium for high mode number, the level spacings in superconducting microwave resonators and, perhaps the most impressive, in the level spacing of a quartz resonator. Mark Oxborrow and his group were able to show the switchover from Poisson to Wigner statistics in a quartz resonator where they started with an integrable geometry and made small perturbations (shaving on a tiny bit of one corner). In their more recent work they were able to correctly identify something like 100k modes for their calculation.[2] To see this many modes AND correctly identify them is a real tour de force. It requires extremely high Q resonators and an accurate way to model them. They also measured parametric correlations predicted by random matrix theory.
Here is a photograph (not simulation) of the Wada basins in the 4 ball scattering problem (from: https://www.nature.com/articles/20573).
[2]: Measurement of parametric correlations in spectra of resonating quartz blocks P Bertelsen, C Ellegaard, T Guhr, M Oxborrow… - Physical Review Letters, 1999