  Radon measures belonging to the negative Sobolev space $H^{-1}(\mathbb{R}^2)$
are important from the point of view of fluid mechanics as they model vorticity
of vortex-sheet solutions of incompressible Euler equations. In this note we
discuss regularity conditions sufficient for nonnegative Radon measures
supported on a line to be in $H^{-1}(\mathbb{R}^2)$. Applying the obtained
results, we derive consequences for measures on $\mathbb{R}^2$ with arbitrary
support and prove elementarily, among other things, that measures belonging to
$H^{-1}(\mathbb{R}^2)$ may be supported on a set of Hausdorff dimension $0$. We
comment on possible numerical applications.
