We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. By using a Fourier multiplier approach, we construct energy functionals to deduce stability in a perturbative setting. For sufficiently high spherical harmonics, we can handle any circular flows with power law singularities or zeros as r ↓ 0 or r ↑ ∞, while for low frequencies we can treat circular flows close to the Taylor–Couette flow. Similar results apply in the planar shear flow case close to Couette.