The shoaling of directionally spread surface gravity waves on a gently sloping beach with straight and parallel depth contours is examined with weakly dispersive Boussinesq theory. In this second-order theory, energy is transferred from the incident waves to components with both higher and lower frequencies in near-resonant nonlinear triad interactions. Directional spreading of the incident waves causes a weak detuning from resonance that is of the same order as the detuning owing to dispersion. Boussinesq theory predictions of the evolution of a single triad (i.e., two primary wave components shoaling from deep water forcing a secondary wave component) are compared to predictions of dispersive finite depth theory for a typical range of beach slopes, incident wave amplitudes, frequencies, and propagation directions. The dependencies of the predicted secondary wave growth on primary wave incidence angles are in good agreement. Whereas the sum frequency response is insensitive to the (deep water) spreading angle of the primary waves, the difference frequency (infragravity) response is significantly reduced for large spreading angles. A stochastic formulation of Boussinesq wave shoaling evolution equations is derived on the basis of the closure hypothesis that phase coupling between quartets of wave components is weak. In this approximation the second- and third-order statistics of random, directionally spread shoaling waves are described by a coupled set of evolution equations for the frequency alongshore wavenumber spectrum and bispectrum. It is shown that a smooth overlap with solutions of dispersive finite depth theory exists in the limit of small beach slope and weak nonlinearity.