  We study ergodic Schr\"odinger operators defined over product dynamical
systems in which one factor is periodic and the other factor is either a
subshift over a finite alphabet or an irrational rotation of the circle. In the
case in which one factor is a Boshernitzan subshift, we prove that either the
resulting operators are periodic or the resulting spectra must be Cantor sets.
The main ingredient is a suitable stability result for Boshernitzan's criterion
under taking products. We also discuss the stability of purely singular
continuous spectrum, which, given the zero-measure spectrum result, amounts to
stability results for eigenvalue exclusion. In particular, we examine
situations in which the existing criteria for the exclusion of eigenvalues are
stable under periodic perturbations. As a highlight of this, we show that any
simple Toeplitz subshift over a binary alphabet exhibits uniform absence of
eigenvalues on the hull for any periodic perturbation whose period is
commensurate with the coding sequence. In the case of a full shift, we give an
effective criterion to compute exactly the spectrum of a random Anderson model
perturbed by a potential of period two, and we further show that the naive
generalization of this criterion does not hold for period three. Next, we
consider quasi-periodic potentials with potentials generated by trigonometric
polynomials with periodic background. We show that the quasiperiodic cocycle
induced by passing to blocks of period length is subcritical when the coupling
constant is small and supercritical when the coupling constant is large. Thus,
the spectral type is absolutely continuous for small coupling and pure point
(for a.e.\ frequency and phase) when the coupling is large.
