Let $G$ be a group with undecidable domino problem (such as $\mathbb{Z}^2$). We prove the undecidability of all nontrivial dynamical properties for sofic $G$-subshifts, that such a result fails for SFTs, and an undecidability result for dynamical properties of $G$-SFTs similar to the Adian-Rabin theorem. For $G$ amenable we prove that topological entropy is not computable from presentations of SFTs, and a more general result for dynamical invariants taking values in partially ordered sets.

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