The widespread prevalence of data breaches amplifies the importance of auditing storage systems. In this work, 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 precisely 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 storage objects. Our formulation considers loggable read-write registers that securely store data using information dispersal and support fast reads. 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 (1) auditability is impossible if $τ\leq 2f $; (2) implementing a weak form of auditability requires $τ\geq 3f+1$; and (3) a stronger form of auditability is impossible. We also show that signing read requests overcomes the lower bound of weak auditability, while totally ordering operations or using non-fast reads enables strong auditability.