  We study theoretically second-order topological superconductors characterized
by the presence of pairs of zero-energy Majorana corner states. We uncover a
quadrupole spin polarization at the system edges that provides a striking
signature to identify topological phases, thereby complementing standard
approaches based on zero-bias conductance peaks due to Majorana corner states.
We consider two different classes of second-order topological superconductors
with broken time-reversal symmetry and show that both classes are characterized
by a quadrupolar structure of the spin polarization that disappears as the
system passes through the topological phase transition. This feature can be
accessed experimentally using spin-polarized scanning tunneling microscopes. We
study different models hosting second-order topological phases, both
analytically and numerically, and using Keldysh techniques we provide numerical
simulations of the spin-polarized currents probed by scanning tips.
