  Consider the time-harmonic acoustic scattering from a bounded penetrable
obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed
to possess a circular conic point or an edge point on the boundary in three
dimensions and a planar corner point in two dimensions. The opening angles of
cones and edges are allowed to be any number in $(0,2\pi)\backslash\{\pi\}$. We
prove that such an obstacle scatters any incoming wave non-trivially (i.e., the
far field patterns cannot vanish identically), leading to the absence of real
non-scattering wavenumbers. Local and global uniqueness results for the inverse
problem of recovering the shape of a penetrable scatterers are also obtained
using a single incoming wave. Our approach relies on the singularity analysis
of the inhomogeneous Laplace equation in a cone.
