A description of the spectral and angular distributions of Compton scattered light in collisions of intense laser pulses with high-energy electrons is unwieldy and usually requires numerical simulations. However, due to the large number of parameters affecting the spectra such numerical investigations can become computationally expensive. Using methods of catastrophe theory we predict higher-dimensional caustics in the spectra of the Compton scattered light, which are associated with bright narrow-band spectral lines, and in the simplest case can be controlled by the value of the linear chirp of the pulse. These findings require no full-scale calculations and have direct consequences for the photon yield enhancement of future nonlinear Compton scattering x-ray or gamma-ray sources.