  Assume that a bounded scatterer is embedded into an infinite homogeneous
isotropic background medium in two dimensions. The refractive index function is
supposed to be piecewise constant. If the scattering interface contains a
weakly or strongly singular point, we prove that the scattered field cannot
vanish identically. This particularly leads to the absence of non-scattering
energies for piecewise analytic interfaces with a weakly singular point. Local
uniqueness is obtained for shape identification problems in inverse medium
scattering with a single far-field pattern.
