  Electromagnetic multipoles have been broadly adopted as a fundamental
language throughout photonics, of which general features such as radiation
patterns and polarization distributions are generically known, while their
singularities and topological properties have mostly slipped into oblivion.
Here we map all the singularities of multipolar radiations of different orders,
identify their indexes, and show explicitly the index sum over the entire
momentum sphere is always $2$, consistent with the Poincar\'{e}-Hopf theorem.
Upon those revealed properties, we can attribute the formation of bound states
in the continuum to overlapping of multipolar singularities with open radiation
channels. This insight unveils a subtle connection between indexes of
multipolar singularities and topological charges of those bound states,
revealing that essentially they are the same. Our work has fused two
fundamental and sweeping concepts of multipoles and topologies, which can
potentially bring unforeseen opportunities for many multipole related fields
within and beyond photonics.
