  The main objective of this paper is to answer the questions posed by Robinson
and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes
equations. Firstly, we prove that the upper box dimension of the potential
singular points set $\mathcal{S}$ of suitable weak solution $u$ belonging in $
L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for $1\leq\frac{2}{q}+\frac{
3}{p}\leq\frac32$ with $2\leq q<\infty$ and $2<p<\infty$ is at most
$\max\{p,q\}(\frac{2}{q}+\frac{ 3}{p}-1)$ in this system. Secondly, it is shown
that $1-2 s$ dimension Hausdorff measure of potential singular points set of
suitable weak solutions satisfying $ u\in
L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac12$ is zero,
whose proof relies on Caffarelli-Silvestre's extension. Inspired by
Baker-Wang's recent work [1], this further allows us to discuss the Hausdorff
dimension of potential singular points set of suitable weak solutions if the
gradient of the velocity under some supercritical regularity.
