We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation $$ H_\epsilon\ \psi \equiv\ \left(\ -\partial_x^2+V_0(x)+q\left(x,x/\epsilon\right)\ \right)\psi=k^2\psi$$ for $k\in\mathbb{R}$ and $\epsilon\ll 1$. Here, $q(\cdot, y+1)=q(\cdot,y)$, has mean zero and $|V_0(x)+q(x,\cdot)|\to 0$ as $|x|\to\infty$. The distorted plane waves of $H_\epsilon$ are solutions of the form $e_{V^\epsilon\pm}(x;k) = e^{\pm ikx}+u^s_\pm(x;k)$, $u^s_\pm$ outgoing as $|x|\to\infty$. We derive their $\epsilon$ small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, $t^\epsilon(k)$, follow. Let $t_0^{hom}(k)$ denote the homogenized transmission coefficient associated with the average potential $V_0$. If the potential is smooth, then classical homogenization theory gives asymptotic expansions of, for example, distorted plane waves, and transmission and reflection coefficients. Singularities of $V_0$ or discontinuities of $q_\epsilon$ are ``interfaces’’ across which a solution must satisfy interface conditions (continuity or jump conditions). To satisfy these conditions it is necessary to introduce interface correctors, which are highly oscillatory in $\epsilon$. Our theory admits potentials which have discontinuities in the microstructure, $q_\epsilon(x)$ as well as strong singularities in the background potential, $V_0(x)$. A consequence of our main results is that $t^\epsilon(k)-t_0^{hom}(k)$, the error in the homogenized transmission coefficient is (i) ${\mathcal O}(\epsilon^2)$ if $q_\epsilon$ is continuous and (ii) ${\mathcal O}(\epsilon)$ if $q_\epsilon$ has discontinuities. Moreover, in the discontinuous case the correctors are highly oscillatory in $\epsilon$, i.e. $\sim \exp({2\pi i\frac{\nu}{\epsilon}})$, for $\epsilon\ll1$. Thus a first order corrector is not well-defined since $\epsilon^{-1}\left(t^\epsilon(k)-t_0^{hom}(k)\right)$ does not have a limit as $\epsilon\to0$. This expression may have limits which depend on the particular sequence through which $\epsilon$ tends to zero. The analysis is based on a (pre-conditioned) Lippman-Schwinger equation, introduced by S.E. Golowich and M.I. Weinstein [Multiscale Model. Simul., 3 (2005), pp. 477–521].