Abstract In this article we are interested in the regularity properties of the probability measure induced by the solution process by a Levy process or a fractional Brownian motion driven Navier–Stokes equations on the two-dimensional torus T . We mainly investigate under which conditions on the characteristic measure of the Levy process or the Hurst parameter of the fractal Brownian motion the law of the projection of u ( t ) onto any finite dimensional F ⊂ L 2 ( T ) is absolutely continuous with respect to the Lebesgue measure on F .