  We consider decaying oscillatory perturbations of periodic Schr\"odinger
operators on the half line. More precisely, the perturbations we study satisfy
a generalized bounded variation condition at infinity and an $L^p$ decay
condition. We show that the absolutely continuous spectrum is preserved, and
give bounds on the Hausdorff dimension of the singular part of the resulting
perturbed measure. Under additional assumptions, we instead show that the
singular part embedded in the essential spectrum is contained in an explicit
countable set. Finally, we demonstrate that this explicit countable set is
optimal. That is, for every point in this set there is an open and dense class
of periodic Schr\"odinger operators for which an appropriate perturbation will
result in the spectrum having an embedded eigenvalue at that point.
