  We prove the mixed-norm Sobolev estimates for solutions to both divergence
and non-divergence form time-dependent Stokes systems with unbounded measurable
coefficients having small mean oscillations with respect to the spatial
variable in small cylinders. As a special case, our results imply Caccioppoli's
type estimates for the Stokes systems with variable coefficients. A new
$\epsilon$-regularity criterion for Leray-Hopf weak solutions of Navier-Stokes
equations is also obtained as a consequence of our regularity results, which in
turn implies some borderline cases of the well-known Serrin's regularity
criterion.
