  We investigate the interior nodal sets $\mathcal{N}_\lambda$ of Steklov
eigenfunctions on connected and compact surfaces with boundary. The optimal
vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular
sets $\mathcal{S}_\lambda$ are finite points on the nodal sets. We are able to
prove that the Hausdorff measure $H^0(\mathcal{S}_\lambda)\leq C\lambda^2$.
Furthermore, we obtain an upper bound for the measure of interior nodal sets
$H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$. Here those positive
constants $C$ depend only on the surfaces.
