We describe a blow-up criterion for the 3D incompressible Euler equations based on inviscid Voigt regularization. Analytical and computational results will be discussed. The 3D Euler-Voigt equations can be though of as a regularization of the 3D Euler equations in the sense that they are globally well-posed, and the solutions approximate the solutions to the 3D Euler equations. Therefore, the blow-up criterion allows one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the "better-behaved" 3D Euler-Voigt equations. We will also discuss a new Voigt-type regularization and blow-up criterion based on the Velocity-Vorticity formulation of the 3D Navier-Stokes equations.