  We define Dirac operators on $\mathbb{S}^3$ (and $\mathbb{R}^3$) with
magnetic fields supported on smooth, oriented links and prove self-adjointness
of certain (natural) extensions. We then analyze their spectral properties and
show, among other things, that these operators have discrete spectrum. Certain
examples, such as circles in $\mathbb{S}^3$, are investigated in detail and we
compute the dimension of the zero-energy eigenspace.
