Bayesian methods such as "stochastic search variable selection" (George and McCulloch 1993 JASA) have been proposed as alternatives to traditional stepwise variable selection procedures in regression models. Instead of either fixing a regression coefficient at zero or allowing it to be estimated by least squares, as in stepwise procedures, stochastic search variable selection posits a mixture prior distribution for the given coefficient, both components being centered at zero but one with a small variance and the other with a large variance. Such a framework lends itself to summaries that average over model uncertainty, such as the posterior probability that a given predictor variable has a non-zero coefficient across the set of main-effect models. But even with complete data, Markov-chain Monte Carlo (MCMC) statistical computing procedures are indicated, and when there are missing covariates, there are choices in the development of a variable-selection procedure. Specifically, one could consider whether to incorporate draws of missing values into the MCMC variable-selection algorithm (an approach we call "simultaneously impute and select", or SIAS) or whether to perform multiple imputation for missing covariates separately and then to run a Bayesian variable-selection algorithm on the multiply imputed data sets (an approach we call "impute, then select", or ITS). In this presentation, we contrast SIAS and ITS with traditional stepwise selection in both simulated settings and in a mental-health services study. We also discuss extensions from models for continuous outcomes to models for binary outcomes.