  We study two relations on multi-dimensional subshifts: A pre-order based on
the patterns configurations contain and the Cantor-Bendixson rank. We exhibit
several structural properties of two-dimensional subshifts: We characterize the
simplest aperiodic configurations in countable SFTs, we give a combinatorial
characterization of uncountable subshifts, we prove that there always exists
configurations without any periodicity but that have the simplest possible
combinatorics in countable SFTs. Finally, we prove that some Cantor-Bendixson
ranks are impossible for countable SFTs, leaving only a few unknown cases.
