  In this paper we study Koszul cohomology and the Green and Prym-Green
conjectures for canonical and Prym-canonical binary curves. We prove that if
property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus
$g$ then it holds for a generic canonical or Prym-canonical binary curve of
genus $g+1$. We also verify the Green and Prym-Green conjectures for generic
canonical and Prym-canonical binary curves of low genus ($6\leq g\leq 15$,
$g\neq 8$ for Prym-canonical and $3\leq g\leq 12$ for canonical).
