Smooth equivalence relations (see the introduction of this paper) are, when viewed up to Borel reducibility, rather boring: a smooth equivalence relation is determined up to Borel reducibility by how many classes it has, and that in turn is always finite, countable, or continuum.
However, the situation seems more interesting when we restrict attention to continuous reducibility. For example, unless I'm missing something it's not hard to show that there is no smooth equivalence relation which is maximal among smooth equivalence relations with respect to continuous reducibility.
I'd like to know more about the structure of smooth equivalence relations with respect to continuous reducibility; what is a good source on this topic? (I'm happy to restrict to equivalence relations on Baire space if that would help, but in general I'm interested in arbitrary Polish spaces.)