  This article concerns the long-term random dynamics in regular spaces for a
non-autonomous Navier-Stokes equation defined on a bounded smooth domain
$\mathcal{O}$ driven by multiplicative and additive noise. For the two kinds of
noise driven equations, we demonstrate the existence of a unique pullback
attractor which is backward compact and asymptotically autonomous in
$\mathbb{L}^2(\mathcal{O})$ and $\mathbb{H}_0^1(\mathcal{O})$, respectively.
The backward-uniform flattening property of the solution is used to prove the
backward-uniform pullback asymptotic compactness of the non-autonomous random
dynamical systems in the regular space $\mathbb{H}_0^1(\mathcal{O})$.
