  We present constructions of countable two-dimensional subshifts of finite
type (SFTs) with interesting properties. Our main focus is on properties of the
topological derivatives and subpattern posets of these objects. We present a
countable SFT whose iterated derivatives are maximally complex from the
computational point of view, constructions of countable SFTs with high
Cantor-Bendixson ranks, a countable SFT whose subpattern poset contains an
infinite descending chain and a countable SFT whose subpattern poset contains
all finite posets. When possible, we make these constructions deterministic,
and ensure the sets of rows are very simple as one-dimensional subshifts.
