Solutions to the Navier-Stokes have been studied in a variety of function spaces. Two important Lebesgue spaces are L2 and L3. In the former, weak solutions exist with monotonic L2 norm, but the equations appear to be ill-posed, and singularities have not been ruled out. In L3, on the other hand, regularity and well-posedness results are available. Weak solutions have been constructed in spaces nearby these, beginning with the work of C. Calderon and continuing with the work of Barker, Sverak and Seregin, among others. The most recent of these focus on spaces with the same scaling as L3, which are said to be critical. In this talk, we will extend this construction from the critical setting to the supercritical setting, filling in a range of spaces between L2 and L3. We will then discuss two applications. The first quantifies how rapidly two hypothetically non-unique solutions evolving from a supercritical initial datum can separate while the second establishes intermediate orders of time-regularity at a singular time and at points away from the singularity.