We consider the problem of capsize of a rolling ship in regular or random beam seas. We use a commonly used model of the motion in which the roll angle is described by a nonlinear oscillator with linear and quadratic damping and harmonic or random wave excitation. In the latter case, we also take into account the hydrodynamic memory effect. The coefficients in the equation are obtained as high-order approximations of real ship data. We calculate numerically capsize criteria in terms of the forcing parameters using Melnikov's asymptotic method and compare with numerical simulations of safe basins for random realisations of the wave excitation obtained from an ocean wave spectrum.