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We consider the structure of aperiodic points in $\mathbb Z^2$-subshifts, andin particular the positions at which they fail to be periodic. We prove that ifa $\mathbb Z^2$-subshift contains points whose smallest period is arbitrarilylarge, then it contains an aperiodic point. This lets us characterise thecomputational difficulty of deciding if an $\mathbb Z^2$-subshift of finitetype contains an aperiodic point. Another consequence is that $\mathbbZ^2$-subshifts with no aperiodic point have a very strong dynamical structureand 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.

Aperiodic points in $\mathbb Z^2$-subshifts