  Using full-wave numerical simulations and analytical multipole expansions we
investigated the properties of real-space polarization singularities that
emerge in light scattering by subwavelength particles. We considered spherical
and torus particles made of dielectric or perfect-electric-conductor material.
We determined the topological indices and the trajectories of electric-field
polarization singularities in both the near-field and far-field regions. In the
far-field region, a total of four singularities are identified and the sum of
their polarization topological indices is two, independent of the particle's
geometric shape. In the near-field region, the polarization singularities
strongly depend on the particle's shape and the polarization of incident light,
and their index sum is not governed by the Poincar\'e-Hopf theorem anymore due
to the non-transverse nature of the fields. From near field to far field, a
flipping of sign can happen to the polarization topological indices of the C
lines. The far-field properties of the singularities can be well explained by
the interference of the excited multipoles, but their near-field properties can
be strongly affected by the evanescent fields that are not captured by the
multipole expansions. Our work uncovers the important relationship between
particles' geometric properties and the polarization singularities of their
scattering field. The results can be applied to manipulate polarization
singularities in nanophotonic systems and could generate novel applications in
optical sensing.
