In the logic based approach to security protocol verification, algorithms for verifying an intruder's knowledge are critical. In this context, the capabilities of an intruder are specified by an equational theory, possibly expressed by a term rewrite system. Previous results have developed algorithms for a number of knowledge problems in many different equational and rewrite theories, such as subterm-convergent. Permutative theories such Associative-Commutative (AC) are of great interest with several procedures having been developed for AC. This leads to the question of decidability of the knowledge problems of deduction and static equivalence in permutative theories in general. It was recently shown that deduction is decidable in permutative theories. However, the decidability of static equivalence (and the related frame distinguishability problem) was still open. In this paper we show that static equivalence is undecidable in permutative theories. In addition, we show that static equivalence remains undecidable in the more restrictive case of leaf permutative theories. On the positive side, static equivalence becomes decidable for a further restricted form of permutative theories we define here.