If X is a deformation space of p-adic Galois representations (either local or global), and S is a set of types, then one can consider the subset X(S) of points in the rigid analytic generic fibre of X whose associated Galois representations are potentially Barsotti--Tate at p of type lying in the set S. We give conditions on the set S which imply that the set X(S) is Zariski dense in X. As one application we conclude that the set of representations which are both potentially Barsotti--Tate and crystabelline at p is Zariski dense in X. This is joint work with Vytautus Paskunas.