  We study the computational and structural aspects of countable
two-dimensional SFTs and other subshifts. Our main focus is on the topological
derivatives and subpattern posets of these objects, and our main results are
constructions of two-dimensional countable subshifts with interesting
properties. We present an SFT whose iterated derivatives are maximally complex
from the computational point of view, a sofic shift whose subpattern poset
contains an infinite descending chain, a family of SFTs whose finite subpattern
posets contain arbitrary finite posets, and a natural example of an SFT with
infinite Cantor-Bendixon rank.
