We study the boundary stabilisation of the wave equation by a nonlinear feedback active on a part of the boundary in geometric situations for which the solutions have singularities. These singularities appear at the interfaces at which the mixed Neumann-Dirichlet boundary conditions meet. Under a simple geometrical condition concerning the orientation of the boundary, we obtain sharp energy decay rates under a general growth assumption on the feedback. We show that the singularities do not affect the energy decay rates and give examples.