There is interest in artificial intelligence for principled techniques to analyze inconsistent information. This stems from the recognition that the dichotomy between consistent and inconsistent sets of formulae that comes from classical logics is not sufficient for describing inconsistent information. We review some existing proposals and make new proposals for measures of inconsistency and measures of information, and then prove that they are all pairwise incompatible. This shows that the notion of inconsistency is a multi-dimensional concept where different measures provide different insights. We then explore relationships between measures of inconsistency and measures of information in terms of the trade-offs they identify when using them to guide resolution of inconsistency.