  In this paper, we intend to reveal how the fractional dissipation
$(-\Delta)^{\alpha}$ affects the regularity of weak solutions to the 3d
generalized Navier-Stokes equations. Precisely, it will be shown that the
$(5-4\alpha)/2\alpha$ dimensional Hausdorff measure of possible time singular
points of weak solutions on the interval $(0,\infty)$ is zero when
$5/6\le\alpha< 5/4$. To this end, the eventual regularity for the weak
solutions is firstly established in the same range of $\alpha$. It is worth
noting that when the dissipation index $\alpha$ varies from $5/6$ to $ 5/4$,
the corresponding Hausdorff dimension is from $1$ to $0$. Hence, it seems that
the Hausdorff dimension obtained is optimal. Our results rely on the fact that
the space $H^{\alpha}$ is the critical space or subcritical space to this
system when $\alpha\geq5/6$.
