In the first part, it is proved that a C2-regular rigid scatterer in R3 can be uniquely
identified by the shear part (i.e. S-part) of the far-field pattern corresponding to all incident shear waves
at any fixed frequency. The proof is short and it is based on a kind of decoupling of the S-part of scattered wave
from its pressure part (i.e. P-part) on the boundary of the scatterer. Moreover, uniqueness using the S-part
of the far-field pattern corresponding to only one incident plane shear wave holds for a ball or a convex
Lipschitz polyhedron. In the second part, we adapt the factorization method to recover the shape of a rigid
body from the scattered S-waves (resp. P-waves) corresponding to all incident plane shear (resp. pressure)
waves. Numerical examples illustrate the accuracy of our reconstruction in R2. In particular, the
factorization method also leads to some uniqueness results for all frequencies excluding possibly a discrete set