  We investigate the stability and softness of nuclei against quadrupole,
octupole, and hexadecapole deformation. By applying the spherical Skyrme-force
Hartree-Fock Bardeen-Cooper-Schrieffer quasi-particle random phase
approximation, we diagnose ground-state deformation when imaginary solutions
are obtained, i.e., the spherical ground state {\em collapses}. We also
calculate the multipole polarizability in spherical nuclei with no collapse, as
a measure of softness. This numerically light and theoretically sound method is
found able to capture deformation patterns across the nuclide chart. The
connection between the intrinsic shape of nuclei and the dynamics of their
low-lying collective states is established and the role of shell structure is
discussed.
