Microwave transport experiments have been performed in a quasi-two-dimensional resonator with randomly distributed scatterers, each mimicking an $r^{-2}$ repulsive potential. Analysis of both stationary wave fields and transient transport shows large deviations from Rayleighs law for the wave height distribution, which can only partially be described by existing multiple-scattering theories. At high frequencies, the flow shows branching structures similar to those observed previously in stationary imaging of electron flow. Semiclassical simulations confirm that caustics in the ray dynamics are likely to be responsible for the observed structures. Particular conspicuous features observed in the stationary patterns are hot spots with intensities far beyond those expected in a random wave field. Reinterpreting the flow patterns as ocean waves in the presence of spatially varying currents or depth variations in the sea floor, the branches and hot spots lead to enhanced frequency of freak or rogue wave formation in these regions.
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