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We initiate the study of auditable storage emulations, which provide the capability for an auditor to report the previously executed reads in a register. We define the notion of auditable register and its properties, and establish tight bounds and impossibility results for auditable storage emulations in the presence of faulty base storage objects. Our formulation considers registers that securely store data using information dispersal (each base object stores only a block of the written value) and supporting fast reads (that complete in one communication round-trip). In such a scenario, given a maximum number f of faulty storage objects and a minimum number τ of data blocks required to recover a stored value, we prove that (R1) auditability is impossible if τ ≤ 2f; (R2) implementing a weak form of auditability requires τ ≥ 3f+1; and (R3) a stronger form of auditability is impossible. We also show that (R4) signing read requests generically overcomes the lower bound of weak auditability, while (R5 and R6) totally ordering operations or using non-fast reads enables strong auditability. These results establish that practical storage emulations need f to 2f additional objects compared to their original lower bounds to support auditability.