The singularity formation problem is a central question in fluid dynamics, and it is still widely open for fundamental models such as the 3d incompressible Euler equations and the Navier-Stokes equations. In this talk, I will first review the singularity formation problem, and I will describe how particle transport poses the main challenge in constructing blow-up solutions — an effect known as “regularization by transport.” I will then outline a new mechanism, arising from the classical Taylor-Couette instability, allowing us to overcome regularization by transport in the 3d Euler equations, thereby constructing the first swirl-driven singularity in R^3. This is joint work with Tarek Elgindi (Duke University).