We describe recent results on studying the dynamics of fluid equations in critical spaces. While it is known that the incompressible Euler equation is ill-posed in the class of Lipschitz velocity fields (even when the data is taken to be smooth away from the origin), we prove well-posedness (global in 2d and local in 3d) for merely Lipschitz data which is smooth away from the origin and satisfies a mild symmetry assumption. To do this requires a deep understanding of the nature of unboundedness of singular integrals on $L^\infty$. After this, we extract a simplified equation which is satisfied by "scale invariant" solutions which lie within the setting of our local well-posedness theory. These scale-invariant solutions, in the 2d Euler setting, can be shown to have very interesting dynamical properties. Moreover, these scale-invariant solutions (while having infinite energy) can be used to prove the existence of finite-energy solutions with the "same" dynamical properties. The talk is based on joint work with I. Jeong.