We discuss some resent results for two type of spectral inequalities in connection with the Navier-Stokes equations. Namely, the Berezin-Li-Yau inequalities for the eigenvalues of elliptic equations and systems ( including the Stokes system) with constant coefficients and the Lieb-Thirring inequalities for the negative spectrum of the Schrodinger operators. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues.