We present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a Hamiltonian quantum system defined by two coupled quartic oscillators, which on the classical level exhibits mixed phase space dynamics. We show that the signature of a pitchfork bifurcation is two-fold: beside the known effect of an enhanced periodic orbit contribution due to its peculiar h-dependence at the bifurcation, we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation. This requires a semiclassical treatment beyond the so-called diagonal approximation. Our semiclassical predictions for both the coarse-grained density of states and the spectral rigidity, are in excellent agreement with corresponding quantum-mechanical results.