100% agree with Nathan's criteria.
If even after those criteria you have more than 1 qualified applicant, you have a few other options, as long as you apply the same requirements equally to everyone.
I verify employment, and call previous landlords for references.
If on the showing the tenant smells like smoke and you don't want smokers in your house, you could pass over somebody for that.
If somebody's car is trashed and filled with garbage, that may be a representation of how they'll treat your unit. You may choose to take a pass on them for that reason.
Brandon Turner in some episodes even mentioned that you could reject an applicant because they have tattoos, if you so choose, as long as you apply that criteria evenly across all applicants.
With that said, tenants should make 3x rent, a 'healthy' credit score, and nothing in collections, judgements, evictions, or criminal history. (I do make exceptions for collections issues in the event of short-term medical issues, where the tenant appears to be on the rebound, but would require a co-signer).