  Codes with various kinds of decipherability, weaker than the usual unique
decipherability, have been studied since multiset decipherability was
introduced in mid-1980s. We consider decipherability of directed figure codes,
where directed figures are defined as labelled polyominoes with designated
start and end points, equipped with catenation operation that may use a merging
function to resolve possible conflicts. This is one of possible extensions
generalizing words and variable-length codes to planar structures. Here,
verification whether a given set is a code is no longer decidable in general.
We study the decidability status of figure codes depending on catenation type
(with or without a merging function), decipherability kind (unique, multiset,
set or numeric) and code geometry (several classes determined by relative
positions of start and end points of figures). We give decidability or
undecidability proofs in all but two cases that remain open.
