  In this paper, we study time-harmonic electromagnetic scattering in two
scenarios, where the anomalous scatterer is either a pair of electromagnetic
sources or an inhomogeneous medium, both with compact supports. We are mainly
concerned with the geometrical inverse scattering problem of recovering the
support of the scatterer, independent of its physical contents, by a single
far-field measurement. It is assumed that the support of the scatterer
(locally) possesses a conical singularity. We establish a local
characterisation of the scatterer when invisibility/transparency occurs,
showing that its characteristic parameters must vanish locally around the
conical point. Using this characterisation, we establish several local and
global uniqueness results for the aforementioned inverse scattering problems,
showing that visibility must imply unique recovery. In the process, we also
establish the local vanishing property of the electromagnetic transmission
eigenfunctions around a conical point under the H\"older regularity or a
regularity condition in terms of Herglotz approximation.
