Relative rigidities
Don't miss Brian Weatherson's very insightful answer to my posting on rigidity (from which I've just stripped some irrelevant formalities). I happily agree with everything he says, so I'll just add a footnote here.
Many advantages of the counterpart theory derive from its denial of the equivalence between 'a=b', 'possibly a=b', and 'necessarily a=b'. For example, this allows for a statue to be identical to a lump of gold even though it might not have been. Since, as Weatherson argues, the rejected equivalence is built into the customary ('strong') concept of rigidity, that concept must be weakened to be useful for counterpart-theorists.
An interesting feature of the weakened concept is that it inevitably inherits the vagueness and context-dependence of 'counterpart': If t rigidly denotes a, its denotation in other worlds depends on contextual factors, in particular, on the term t itself. E.g. 'the (actual) lump' and 'the (actual) statue' rigidly denote the same object, but the rigidity is of a different kind, as not every counterpart of the lump is a counterpart of the statue. In using 'the statue', we invoke a statue-counterpart relation, whereas in using 'the lump', we invoke a lump-counterpart relation. Likeweise, 'rigid' should be indexed: 'the (actual) lump' lump-rigidly denotes the statue. This is unfamiliar, but may actually be clearer than the old concept of rigidity. Instead of stipulating that 'water' rigidly denotes water, and 'Hesperus' rigidly denotes Hesperus, we can now stipulate that 'water' substance-rigidly denotes water, and 'Hesperus' planet-rigidly denotes Hesperus, thereby stating explicitly that an otherworldy intrinsic duplicate of Hesperus that is part of a big mountain and hence not a planet, is to be excluded.
However, the difference between rigid and non-rigid designators becomes somewhat blurred. For we can also stipulate that 'water' phenomenally-rigidly denotes water (the relevant counterparts are counterparts by acquaintance, as in Lewis, 'Individuation by acquaintance and by stipulation'), or that 'pain' functionally-rigidly denotes pain.