A powerful technique originally introduced in the regularity theory of minimal surfaces is to `zoom in' on a potential `singularity' and classify the `blow-up limit'. We discuss applications of the above technique to the Navier-Stokes equations. In particular, when `zooming in' on a potential Navier-Stokes singularity, sequences of Navier-Stokes solutions whose initial data are converging only in a weak-* sense arise naturally. We identify a class of solutions satisfying the following stability property: weak-* convergence of the initial data in `critical spaces' implies strong convergence of the corresponding solutions. We apply the weak-* stability property to problems concerning blow-up criteria in critical spaces, minimal blow-up data, and forward self-similar solutions. Joint work with Tobias Barker (ENS).