  In this paper, we are concerned with the precise relationship between the
Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable
weak solutions and the parameter $\alpha$ in the nonlinear term in the
following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^\alpha=f.$$ It
is shown that when $5/3\leq\alpha<7/3$, the
$\frac{3\alpha-5}{\alpha-1}$-dimensional parabolic Hausdorff measure of
$\mathcal{S}$ is zero, which generalizes the recent corresponding work of
Oz\'anski and Robinson in [31,SIAM J. Math. Anal. 51: 228--255, 2019] for
$\alpha=2$ and $f=0$. The same result is valid for a 3D modified Navier-Stokes
system.
