We study the problem of augmenting relational tuples with inconsistency awareness and tackling top-k queries under a set of denial constraints (DCs). We define a notion of inconsistent tuples with respect to a set of DCs and define two measures of inconsistency degrees, which consider single and multiple violations of constraints. In order to compute these measures, we leverage two models of provenance, namely why-provenance and provenance polynomials. We investigate top-k queries that allow to rank the answer tuples by their inconsistency degrees. Since one of our measure is monotonic and the other non-monotonic, we design an integrated top-k algorithm to compute the top-k results of a query w.r.t. both inconsistency measures. By means of an extensive experimental study, we gauge the effectiveness of inconsistency-aware query answering and the efficiency of our algorithm with respect to a baseline, where query results are fully computed and ranked afterwards.