  We study the space of continuous $Z^d$-actions on the Cantor set,
particularly questions on the existence and nature of actions whose isomorphism
class is dense (Rohlin's property). Kechris and Rosendal showed that for $d=1$
there is an action on the Cantor set whose isomorphism class is residual. We
prove in contrast that for $\geq 2$ every isomorphism class is meager; on the
other hand, while generically an action has dense isomorphism class and the
effective actions are dense, no effective action has dense isomorphism class.
Thus for $d \geq 2$ conjugation on the space of actions is topologically
transitive but one cannot construct a transitive point. Finally, we show that
in the space of transitive and minimal actions the effective actions are
nowhere dense, and in particular there are minimal actions that are not
approximable by minimal SFTs.
