Radiations of large clades often accompany rapid morphological diversification. Evolutionary biologists debate the impact of external restrictions imposed by ecology, and intrinsic constraints imposed by development and genetics, on the rate at which morphological innovations are gained. These issues are particularly interesting for groups such as tetrapods, which evolved novel body plans relative to their piscine ancestors and which also invaded new ecosystems following terrestrialization. Prior studies have addressed these issues by looking at either ranges of morphological variation or rates of character change. Here, we address a related but distinct issue: the numbers of characters that freely vary within a clade. We modify techniques similar to those used by ecologists to infer species richnesses to estimate the number of potentially varying characters given the distributions of changes implied by a model phylogeny. Our results suggest both increasing constraints/restrictions and episodes of 'character release' (i.e. increasing the number of potentially varying characters). In particular, we show that stem lissamphibians had a restricted character space relative to that of stem amniotes, and that stem amniotes both had restrictions on some parts of character space but also invaded new character space that had been unavailable to stem tetrapods.