We consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination and we prove several results in this new research area. First, we apply the asymptotic distinguishability criterion of [Kleinmann et al. Phys. Rev. A 84 042326 (2011)] to the classic double-trine ensemble and resolve a long-standing conjecture concerning the sub-optimality of LOCC. We then consider N copies of the trine states and establish a tight connection between the N-copy ensemble and Shor’s lifted single-copy ensemble. For any finite N, we prove that optimal identification of the states cannot be achieved by LOCC; however as N → ∞, LOCC can indeed discriminate the states optimally. Finally, we turn to the subject of distinguishability norms and we derive a new necessary condition for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this general result to prove the first known gap between the LOCC and Separable norms.