The logic of bunched implications BI provides a logical analysis of a basic notion of resource rich enough to provide a ``pointer logic'' semantics for programs which manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, the challenge consists in dealing with BI's Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. As consequences, we have two strong new results for BI: the decidability of propositional BI and the finite model property with respect to Grothendieck topological semantics. In addition, we propose a new semantics which generalizes the semantics of BI's pointer logic and for which BI is complete.