Lagrangian submanifolds of symplectic manifolds are very central
objects in classical mechanics and microlocal analysis. These
manifolds are frequently singular (integrable systems, bifurcations,
reduction). There have been many works on singular Lagrangian
manifolds initiated by V. Arnold, A. Givental, and others. The goal of
our paper is to extend the classical and semiclassical normal forms of
completely integrable systems near nondegenerate (Morse-Bott)
singularities to more singular systems. It turns out that there is a
nicely working way to do that, leading to normal forms and universal
unfoldings. We obtain in this way natural ansatzes extending the
Wentzel-Kramers-Brillouin(WKB)-Maslov ansatz. We give more details on
the simplest non-Morse example, the cusp, which corresponds to a
saddle-node bifurcation.