The objective of this study is to assess the stability, especially at and near critical frequencies, of a new unified, stable, symmetric, domain finite element and boundary integral methodology for solving time-harmonic interface problems for scatterers of arbitrary shape. The validity of this energy-based variational procedure for rigid and penetrable smooth scatterers, including the existence, uniqueness and optimal convergence of the corresponding numerical approximations, has been proved rigorously in the context of fluid-structure interaction problems. The emphasis here is on investigating the applicability of this procedure to the case of scatterers with corners, using square and thin rectangular two-dimensional rigid obstacles as prototypes. We also establish that the apparently distinct direct boundary integral formulation due to Burton and Miller and the combined single- and double-layer indirect formulation due to Leis, Brakhage and Werner, and Panich, introduced to insure that boundary integral equations for scattering problems are uniquely solvable for all wavenumbers, are entirely equivalent within our variational setting. By examining the condition number of the matrix of coefficients of the discretized equations, as well as the resulting solutions, both directly on the surface of the scatterers and in the far field, it is demonstrated that the new methodology is robust, and completely insensitive to critical frequencies, even for thin objects. Most significantly, the precise singular behaviour of the pressure field at corners is also predicted quite accurately by using standard finite elements along the boundary of the scatterer, without having to introduce special singularity functions, which for general interface problems, may not be known in advance.