We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindelöf degree, the extent, the Rothberger property, $σ$-compactness and the Menger property for ray, end and edge-end spaces. We show that $σ$-compactness and the Menger property are equivalent for these spaces, and that they are all $D$-spaces. As an application of some of these characterizations, we are able to provide combinatorial characterizations of graphs with countably many ends and edge-ends.