Abstract: We study the eigenpairs of the Dirichlet Laplacian for plane waveguides with corners. We prove that in presence of a non-trivial corner there exist eigenvalues under the essential spectrum. Moreover we provide accurate asymptotics for eigenpairs associated with the lowest eigenvalues in the small angle limit. For this, we also investigate the eigenpairs of a one-dimensional toy model related to the Born-Oppenheimer approximation, and of the Dirichlet Laplacian on triangles with sharp angles. This is joint work with Monique Dauge.