In the presentation we will discuss our research program concerning the study of extreme vortex events in viscous incompressible flows. These vortex states arise as the flows saturating certain fundamental mathematical estimates, such as the bounds on the maximum enstrophy growth in 3D (Lu & Doering, 2008). They are therefore intimately related to the question of singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it allows one to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of new results concerning 2D and 3D flows characterized by the maximum possible growth of, respectively, palinstrophy and enstrophy. It will be shown that certain types of initial data, such as the Taylor-Green vortex, which have been used in numerous computational studies of the blow-up problem are in fact a particular instance (corresponding to an asymptotic limit) of our family of extreme vortex states. We will present results comparing the growth of relevant quantities in high-resolution direct numerical simulations of the Navier-Stokes system obtained using our extreme vortex states and different initial data employed in earlier studies.