We investigate the zero modes for three-dimensional Dirac operators with singular magnetic fields supported on links. They can be seen as a generalization of Aharonov-Bohm solenoids, in particular they exhibit the same $2\pi$-periodicity of the fluxes carried by the field lines. The occurrence of zero modes is studied through the spectral flow of loops of such singular operators: it is generically non-zero and depends on the geometry of the field lines (not only their topology). This a joint work with Fabian Portmann and Jan Philip Solovej.