  It is known that the energy of a weak solution to the Euler equation is
conserved if it is slightly more regular than the Besov space
$B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to)
a smooth manifold, we derive various $L^p$-space regularity criteria
dimensionally equivalent to the critical one. In particular, if the singular
set is a hypersurface the energy of $u$ is conserved provided the one sided
non-tangential limits to the surface exist and the non-tangential maximal
function is $L^3$ integrable, while the maximal function of the pressure is
$L^{3/2}$ integrable. The results directly apply to prove energy conservation
of the classical vortex sheets in both 2D and 3D at least in those cases where
the energy is finite.
