We propose a new highly flexible and tractable Bayesian approach to undertake variable selection in non-Gaussian regression models. It uses a copula decomposition for the joint distribution of observations on the dependent variable. This allows the marginal distribution of the dependent variable to be calibrated accurately using a nonparametric or other estimator. The family of copulas employed are "implicit copulas" that are constructed from existing hierarchical Bayesian models widely used for variable selection, and we establish some of their properties. Even though the copulas are high dimensional, they can be estimated efficiently and quickly using Markov chain Monte Carlo. A simulation study shows that when the responses are non-Gaussian, the approach selects variables more accurately than contemporary benchmarks. A real data example in the Web Appendix illustrates that accounting for even mild deviations from normality can lead to a substantial increase in accuracy. To illustrate the full potential of our approach, we extend it to spatial variable selection for fMRI. Using real data, we show our method allows for voxel-specific marginal calibration of the magnetic resonance signal at over 6000 voxels, leading to an increase in the quality of the activation maps.