<p>Strong-field processes are highly nonlinear, often involving the absorption of thousands of photons, which can lead to the emission of high harmonics in the soft X-ray region or the production of high-energy electrons carrying information about the target. The transition amplitudes of these processes are typically expressed as integrals over time, where the phase of the subintegral function is determined by a strong-field-dependent action. This structure allows the use of asymptotic methods for integrals and forms the basis of quantum-orbit theory, which provided valuable physical insight and led to new discoveries. Here, we show how highly nonlinear laser-assisted processes can be systematically investigated using the powerful mathematical apparatus of catastrophe theory. Using laser-assisted scattering as an example, we demonstrate how catastrophe theory can identify critical values of control parameters for which qualitative changes occur in the spectral response, i.e., where the catastrophes appear. We illustrate this with the fold, cusp, swallowtail, and wigwam catastrophes.</p>