We examine vacuum polarization effects around a singular magnetic flux string in 2 + 1 and 3 + 1 dimensions. In particular we determine the local effects: charge and current densities and the flux induced by the vacuum current. The analysis is performed entirely on the basis of exact solutions to the pertinent Dirac equations, and we find that the (3 + 1)-dimensional effects can be described in a simple way as a superposition of the (2 + 1)-dimensional effects.