How should we quantify the amount of inconsistency in the database, when consistency is defined in terms of a given set of integrity constraints or rules? Proper measures are important for various tasks, such as progress indication and action prioritization in cleaning systems, and reliability estimation for new datasets. To choose an appropriate inconsistency measure for a specific use case, it is important to identify the desired properties of the application and understand which of these is guaranteed or, at least, expected in practice. For example, in some use cases, the inconsistency should reduce if constraints are eliminated; in others, it should be stable and avoid jitters and jumps in reaction to small changes in the database. Building on past research on inconsistency measures for knowledge bases, we embark on a systematic investigation of important properties for inconsistency measures. We investigate a collection of basic measures that have been proposed in the past in both the Knowledge Representation and Database communities, analyze their theoretical properties, and empirically observe their behavior in an experimental study. We also demonstrate how the framework can lead to new inconsistency measures by introducing a new measure that, in contrast to the rest, satisfies all of the properties we consider and can be computed in polynomial time.