  In this paper, we study the local well-posedness of Chern-Simons-Dirac system
in the Coulomb gauge for initial data in $H^s(\mathbb R^2)$ for $s>0$. The
novelty of this paper is to prove almost critical regularity by using the
bilinear estimates of wave type localized in a thickened null cone, given by
\cite{selb} via null structure. We also prove the Dirac spinor flow of
Chern-Simons-Dirac system cannot be $C^3$ at the origin in $H^s$ if $s<0$.
