From Smullyan and Fitting's Set Theory and the Continuum Problem:

Which classes are sets? Rather than attempt an absolute answer to this (which some authors have done with dubious success), we regard it as philosophically more honest to take these notions as only relative to any given model of the axioms of class-set theory.

This is fine for the text, but it makes me curious what attempts have been made at absolute answers, dubious or not.