We survey a selection of inconsistency measures from the literature and investigate their computational complexity wrt. decision problems related to bounds on the inconsistency value and the functional problem of determining the actual value. Our findings show that those inconsistency measures can be partitioned into four classes related to their complexity. The first three classes contain measures whose complexities are located on the first three levels of the polynomial hierarchy, respectively. The final class is under standard complexity-theoretic assumptions located beyond the polynomial hierarchy. We provide membership results for all the investigated problems and completeness results for most of them. In addition, we undertake a preliminary study on the computational complexity of the measures on fragments of propositional logic.