In this paper, a new heuristic-based approach is introduced to extract minimally unsatisfiable subformulas (in short, MUSes) of SAT instances. It is shown that it often outperforms current competing methods. Then, the focus is on inconsistent covers, which represent sets of MUSes that cover enough independent sources of infeasibility for the instance to regain satisfiability if they were repaired. As the number of MUSes can be exponential with respect to the size of the instance, it is shown that such a concept is often a viable trade-off since it does not require us to compute all MUSes but provides us with enough mutually independent infeasibility causes that need to be addressed in order to restore satisfiability.