Let P be a second order strictly hyperbolic operator with \(C^{\infty}\) coefficients in some open domain \(\Omega \subset {\mathbb{R}}^{1+n}\). The propagation of conormal regularity for bounded solutions of the (weakly) semilinear hyperbolic equation \[ (*)\quad Pu(z)=g(z,u),\quad z\in \Omega,\quad u\in L^{\infty}_{loc}(\Omega),\quad g\in C^{\infty}(\Omega \times {\mathbb{R}}) \] will be described when the wavefront, a characteristic surface for P, has cusp singularities.