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In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameter α in the nonlinear term in the following parabolic equation h t + h x x x x + ∂ x x | h x | α = f. It is shown that when 5 / 3 ≤ α < 7 / 3 , the 3 α − 5 α − 1 dimensional parabolic Hausdorff measure of S is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal., 51, 228–255 (2019)] for α = 2 and f = 0. The same result is valid for a 3D modified Navier–Stokes system. [ABSTRACT FROM AUTHOR]