  Fast electrons spectromicroscopies enable to measure quantitatively the
optical response of excitations with unrivaled spatial resolution. However, due
to their inherently scalar nature, electron waves cannot access to
polarization-related quantities. In spite of promising attempts based on the
conversion of concepts originating from singular optics (such as vortex beams),
the definition of an optical polarization analogue for fast electrons has
remained a dead letter. Here, we establish such an analogue as the dipole
transition vector of the electron between two well-chosen singular wave states.
We show that electron energy-loss spectroscopy (EELS) allows a direct
measurement of the \textit{polarized} electromagnetic local density of states.
In particular, in the case of circular polarization, it measures directly the
local optical spin density. This work establishes EELS as a quantitative
technique to tackle fundamental issues in nano-optics, such as super-chirality,
the local polarization of dark excitations or polarization singularities at the
nanoscale.
