Real-world databases are often inconsistent. Although there has been an extensive body of work on handling inconsistency, little work has been done on measuring inconsistency in databases. In this paper, building on work done on measuring inconsistency in propositional knowledge bases, we explore inconsistency measures (IMs) for databases with denial constraints. We first introduce new database IMs that are inspired by well-established methods to quantify inconsistency in propositional knowledge bases, but are tailored to the relational database context where data are generally the reason for inconsistency, not the integrity constraints. Then, we analyze the compliance of the database IMs with rationality postulates, and investigate the complexity of the inconsistency measurement problem as well as of the problems of deciding whether the inconsistency is lower than, greater than, or equal to a given threshold.