Extreme waves at the sea surface can have severe impacts on marine structures. One of the theoretical mechanisms leading to extreme waves is the instability of deep-water wave trains subject to initially small perturbations, which then grow exponentially. The present study focuses on the two-dimensional Benjamin–Feir (or modulational) instability and the three-dimensional crescent (or horseshoe) waves, also known as Class I and Class II instabilities, respectively. Numerical studies on Class I and Class II wave instabilities to date have been limited to models founded on potential flow theory, thus they could only properly investigate the process from initial growth of the perturbations to the initial breaking point. The present study conducts numerical simulations to investigate the generation and development of wave instabilities involving the wave breaking process. A CFD model solving Reynolds-averaged Navier-Stokes (RANS) equations coupled with turbulence closure in terms of the anisotropic Reynolds stress model is applied. Wave form evolutions, Fourier amplitudes, and the turbulence beneath the broken waves are investigated.