It is generally difficult to know whether the parameters in nonlinear econometric models are point‐identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of the full parameter vector and of subvectors in models defined through a likelihood or a vector of moment equalities or inequalities. The CSs are based on level sets of “optimal” criterion functions (such as likelihoods, optimally‐weighted or continuously‐updated GMM criterions). The level sets are constructed using cutoffs that are computed via Monte Carlo (MC) simulations from the quasi‐posterior distribution of the criterion. We establish new Bernstein–von Mises (or Bayesian Wilks) type theorems for the quasi‐posterior distributions of the quasi‐likelihood ratio (QLR) and profile QLR in partially‐identified models. These results imply that our MC CSs have exact asymptotic frequentist coverage for identified sets of full parameters and of subvectors in partially‐identified regular models, and have valid but potentially conservative coverage in models whose local tangent spaces are convex cones. Further, our MC CSs for identified sets of subvectors are shown to have exact asymptotic coverage in models with singularities. We provide local power properties and uniform validity of our CSs over classes of DGPs that include point‐ and partially‐identified models. Finally, we present two simulation experiments and two empirical examples: an airline entry game and a model of trade flows.