  We consider two-dimensional Pauli and Dirac operators with a polynomially
vanishing magnetic field. The main results of the paper provide resolvent
expansions of these operators in the vicinity of their thresholds. It is proved
that the nature of these expansions is fully determined by the flux of the
magnetic field. The most important novelty of the proof is a comparison between
the spatial asymptotics of the zero modes obtained in two different manners.
The result of this matching allows to compute explicitly all the singular terms
in the associated resolvent expansions. As an application we show how the
magnetic field influences the time decay of the associated wave-functions
quantifying thereby the paramagnetic and diamagnetic effects of the spin.
