  We compile a complete list of isomorphism class representatives of curves of
genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves
in each stratum of the Brill--Noether stratification of the moduli space
$\mathcal{M}_6$, due to Mukai in the generic case. Our computed value of
$\#\mathcal{M}_6(\mathbb{F}_2)$ agrees with the Lefschetz trace formula as
recently computed by Bergstrom--Canning--Petersen--Schmitt.
