  In this note we study singular oscillatory integrals with linear phase
function over hypersurfaces which may oscillate, and prove estimates of $L^2
\mapsto L^2$ type for the operator, as well as for the corresponding maximal
function. If the hypersurface is flat, we consider a particular class of a
nonlinear phase functions, and apply our analysis to the eigenvalue problem
associated with the Helmholtz equation in $\mathbb{R}^3$.
