There are many things you could do to select variables from multiply imputed data, but not all yield appropriate estimates. See [Wood et al (2008) Stat Med](http://onlinelibrary.wiley.com/doi/10.1002/sim.3177/abstract) for a comparison of various possibilities. 

I have found the following two-step procedure useful in practice.

 1. Apply your preferred variable selection method independently to each of the $m$ imputed data sets. You will end up with $m$ different models. For each variable, count the number of times it appears in the model. Select those variables that appear in at least half of the $m$ models. 
 2. Use the p-value of the Wald statistic or of the likelihood ratio test as calculated from the $m$ multiply-imputed data sets as the criterion for further stepwise model selection.

The pre-selection step 1 is included to reduce the amount of computation. See https://stefvanbuuren.name/fimd/sec-stepwise.html (section 5.4.2) for a code example of the two-step method in R using `mice()`. In Stata, you can perform Step 2 (on all variables) with `mim:stepwise`.