 The study of Compton scattering of X-rays off electrons driven by a relativistically intense short optical laser pulse reveals a broad plateau in the frequency spectrum due to non-linear mixing between X-ray and laser photons, with sharp peak structures in the plateau region highly sensitive to the shape of the laser pulse. These structures are interpreted as spectral caustics using semi-classical analysis of the laser assisted QED matrix element, relating the caustic peak locations to the laser driven electron motion. This article was authored by DCPT, Assertsikov, S Fritzsch and others.