We discuss ideal presentations of effective quasi-Polish spaces and some of their subclasses. Based on this, we introduce and study natural numberings of these classes, in analogy with the numberings of classes of algebraic structures popular in computability theory. We estimate the complexity of (effective) homeomorphism w.r.t. these numberings, and of some natural index sets. In particular, we give precise characterizations of the complexity of certain classes related to separation axioms.