  We investigate $L^1\to L^\infty$ dispersive estimates for the massless two
dimensional Dirac equation with a potential. In particular, we show that the
Dirac evolution satisfies the natural $t^{-\frac12}$ decay rate, which may be
improved to $t^{-\frac12-\gamma}$ for any $0\leq \gamma<\frac{3}{2}$ at the
cost of spatial weights. We classify the structure of threshold obstructions as
being composed of a two dimensional space of p-wave resonances and a finite
dimensional space of eigenfunctions at zero energy. We show that, in the
presence of a threshold resonance, the Dirac evolution satisfies the natural
decay rate except for a finite-rank piece. While in the case of a threshold
eigenvalue only, the natural decay rate is preserved. In both cases we show
that the decay rate may be improved at the cost of spatial weights.
