There have been recent breakthroughs on the study of the incompressible Euler equations at critical regularity, at which one can barely (or barely cannot) close the a priori estimates necessary for local well-posedness. In a series of works with T. Elgindi, we have initiated the study of the Euler equations in critical spaces which contain functions singular only at a single point and smooth otherwise. Our results show that there is local well-posedness if and only if the solution satisfies an appropriate rotational symmetry assumption. We discuss in some detail the relevant theory in the vortex patch case.