We consider the simplest instabilities involving multiple unstable electrostatic plasma waves corresponding to four-dimensional systems of mode amplitude equations. In each case, the coupled amplitude equations are derived up to third-order terms. The nonlinear coefficients are singular in the limit in which the linear growth rates vanish together. These singularities are analysed using techniques developed in previous studies of a single unstable wave. In addition to the singularities familiar from the single-mode problem, there are new singularities in coefficients coupling the modes. The new singularities are most severe when the two waves have the same linear phase velocity and satisfy the spatial resonance condition k2=2k1. As a result, the short-wave mode saturates at a dramatically smaller amplitude than that predicted for the weak-growth-rate regime on the basis of single-mode theory. In contrast, the long-wave mode retains the single-mode scaling. If these resonance conditions are not satisfied then both modes retain their single-mode scaling and saturate at comparable amplitudes.