  We consider the structure of aperiodic points in $\mathbb Z^2$-subshifts, and
in particular the positions at which they fail to be periodic. We prove that if
a $\mathbb Z^2$-subshift contains points whose smallest period is arbitrarily
large, then it contains an aperiodic point. This lets us characterise the
computational difficulty of deciding if an $\mathbb Z^2$-subshift of finite
type contains an aperiodic point. Another consequence is that $\mathbb
Z^2$-subshifts with no aperiodic point have a very strong dynamical structure
and are almost topologically conjugate to some $\mathbb Z$-subshift. Finally,
we use this result to characterize sets of possible slopes of periodicity for
$\mathbb Z^3$-subshifts of finite type.
