Recent estimates of the crustal thickness of Mars show a bimodal result of either ∼20 or ∼40 km beneath the InSight lander. We propose an approach based on random matrix theory applied to receiver functions (RFs) to further constrain the subsurface structure. Assuming a spiked covariance model for our data, we first use the phase transition properties of the singular value spectrum of random matrices to detect coherent arrivals in the waveforms. Examples from terrestrial data show how the method works in different scenarios. We identify three previously undetected converted arrivals in the InSight data, including the first multiple from a deeper third interface. We then use this information to jointly invert RFs with the absolute S-wave velocity information in the polarization of body waves. Results show a crustal thickness of 43 ± 5 km beneath the lander with two mid-crustal interfaces at depths of 8 ± 1 and 21 ± 3 km.