In the search for possible blow-up of the incompressible Navier-Stokes equations, there has been much recent attention on the class of axisymmetric solutions with swirl. Several interesting structures of this system have led to regularity criteria that are unknown for the general system. This talk concerns quantitative regularity theorems which, among other things, can constrain the numerical search for blow-up. For instance, if the solution concentrates down to length scale L in an approximately self-similar way, we prove that the dimensionless "size'' of the profile must be at least (log log (1/L))^c. More generally, all subcritical norms of the solution can be controlled explicitly in terms of the critical weak-L^3 norm. This is joint work with Wojciech Ozanski (FSU).