Global Regularity of the 3D Navier–Stokes Equations

For the incompressible Navier–Stokes equations in three dimensions,
$$\partial_t u + (u\cdot\nabla)u = -\nabla p + \nu\,\Delta u,\qquad \nabla\cdot u = 0,$$
do smooth, globally defined solutions exist for all smooth divergence-free initial data $u_0$ with finite energy — or can a finite-time singularity form?

Establishing global existence and smoothness (or exhibiting a blow-up) would settle the mathematical foundation of turbulence.