Nonlinear Thomson and Compton processes, in which energetic electrons collide with an intense optical pulse, are investigated in the framework of classical and quantum electrodynamics. Spectral modulations of the emitted radiation, appearing as either oscillatory or pulsating structures, are observed and explained. It is shown that both processes generate a bandwidth radiation spanning the range of few MeV, which occurs in a small cone along the propagation direction of the colliding electrons. Most importantly, these broad bandwidth structures are temporarily coherent which proves that Thomson and Compton processes lead to generation of a supercontinuum. It is demonstrated that the radiation from the supercontinuum can be synthesized into zeptosecond (possibly even yoctosecond) pulses. Thus, confirming that Thomson and Compton scattering can be used as novel sources of an ultra-short radiation, opening routes to new physical domains for strong laser physics.