  We analyze the behavior of the ensemble of surface boundaries of the critical
clusters at $T=T_c$ in the $3d$ Ising model. We find that $N_g(A)$, the number
of surfaces of given genus $g$ and fixed area $A$, behaves as $A^{-x(g)}$
$e^{-\mu A}$. We show that $\mu$ is a constant independent of $g$ and $x(g)$ is
approximately a linear function of $g$. The sum of $N_g(A)$ over genus scales
as a power of $A$. We also observe that the volume of the clusters is
proportional to its surface area. We argue that this behavior is typical of a
branching instability for the surfaces, similar to the ones found for
non-critical string theories with $c > 1$. We discuss similar results for the
ordinary spin clusters of the $3d$ Ising model at the minority percolation
point and for $3d$ bond percolation. Finally we check the universality of these
critical properties on the simple cubic lattice and the body centered cubic
lattice.
