This paper deals with the propagation of singularities in thin elastic shells whose middle surface is not of an uniform nature. Numerical computations are performed using an adaptive mesh procedure proposed by the software Abaqus to refine the mesh inside the internal layers. The computation are done on three kinds of shells: hyperbolic-parabolic, hyperbolic-elliptic and parabolic-elliptic. The numerical results enable us to determine the propagation of singularities in such shells and to have information about their nature. In particular, when a singular force is applied in the hyperbolic part of a shell, the numerical computations show that a singularity propagates in the hyperbolic part and that a reflection occurs at the boundary with a part of a different nature (parabolic or elliptic).