  For singular numbers of integral operators of the form $u(x)\mapsto \int
F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)\mu(dY),$ with measure $\mu$ singular with respect to
the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the
counting function are established. The kernel $K(X,Y,Z)$ is supposed to be
smooth in $X,Y$ and in $Z\ne 0$ and to admit an asymptotic expansion in
homogeneous functions in $Z$ variable as $Z\to 0.$ The order in estimates is
determined by the leading homogeneity order in the kernel and geometric
properties of the measure $\mu$ and involves integral norms of the weight
functions $F_1,F_2$. For the case of the measure $\mu$ being the surface
measure for a Lipschitz surface of some positive codimension $\mathfrak{d},$ in
the self-adjoint case, the asymptotics of eigenvalues of this integral operator
is found.
