We describe the Ziegler spectra for a large class of self-injective algebras of polynomial growth representation type. To derive our results, we extend the methods and techniques used in studying the finite-dimensional representation theory of such algebras. In particular, the use of tubular extensions and covering functors, and the reduction into various (e.g. tilting, socle, or stable) equivalence classes. Our broadest result reduces the description of the Ziegler topology, for an arbitrary finite-dimensional self-injective algebra of polynomial growth, to such a description over a tame hereditary or tubular algebra. In the case of trivial extensions, we are able to give a complete description and a construction of the infinite-dimensional points lying in the closure of each Auslander-Reiten component.