  We present a comprehensive report on the relationships between variations of
the Menger and Rothberger selection properties with respect to $\omega$-covers
and $k$-covers in the most general topological setting and address the finite
productivity of some of these properties. We collect various examples that
separate certain properties and we carefully identify which separation axioms
simplify aspects of these properties. We finish with a consolidated list of
open questions focused on topological examples.
