  We investigate numerically the effect of regulating fermions in the presence
of singular background fields in three dimensions. For this, we couple free
lattice fermions to a background compact U(1) gauge field consisting of a
monopole-anti-monopole pair of magnetic charge $\pm Q$ separated by a distance
$s$ in a periodic $L^3$ lattice, and study the low-lying eigenvalues of
different lattice Dirac operators under a continuum limit defined by taking
$L\to\infty$ at fixed $s/L$. As the background gauge field is parity even, we
look for a two-fold degeneracy of the Dirac spectrum that is expected of a
continuum-like Dirac operator. The naive-Dirac operator exhibits such a
parity-doubling, but breaks the degeneracy of the fermion-doubler modes for the
$Q$ lowest eigenvalues in the continuum limit. The Wilson-Dirac operator lifts
the fermion-doublers but breaks the parity-doubling in the $Q$ lowest modes
even in the continuum limit. The overlap-Dirac operator shows parity-doubling
of all the modes even at finite $L$ that is devoid of fermion-doubling, and
singles out as a properly regulated continuum Dirac operator in the presence of
singular gauge field configurations albeit with a peculiar algorithmic issue.
