In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as observability inequalities on space-time measurable sets of positive measure for nonstationary Stokes system. The latter extends the result established recently by Wang and Zhang [SIAM J. Control Optim., 55 (2017), pp. 1862-1886] to the case of observations from subsets of positive measure in both time and space variables. Furthermore, we present their applications in the shape optimization problem, as well as the time optimal control problem for the Stokes system. In particular, we give a positive answer to an open question raised by Privat, Trelat, and Zuazua [Arch. Rational Mech. Anal., 216 (2015), pp. 921-981].