This goes back a few years and some of these practices have stopped...for the most part.
1. Mother of 8 stay at home dad. The irony is that dad was chronically under employed and the family long ago made the decision that mom would work and dad would stay at home. Being a working mom was fine before the woman went back to get her PhD...prior to becoming a professor, she worked full time in BYU admissions. The VPs on campus that did her interview didn't think she would be the right roll model. This case took some serious intervention and was finally overturned.
2. Female new PhD. Husband was a doctor and planned on working in a clinic about a 3 hour drive away in Idaho (where is father and brother owned a practice in this particular specialty). Someone in the admin (VP academics at the time) thought that the professor commuting and having two places of residence was not in keeping with the Proclamation. I had to fight this one hard. At one point, pres Samuelson told the VP of academics to call over the Associate dean of our college so that they could explain their decision. He said: "Steve's a stake president, he'll understand." Steve went over, listened to their explanation and said: Well, you are correct, I am a stake president. But you're wrong. I don't understand, I do not agree, and you're flat out wrong. ...After more fighting and argument over several days, Pres Samuelson consulted with his bosses who told them he was wrong, the family's personal decision about dual career and living arrangements was not a matter of worthiness nor fitness for employment at BYU.

Cases of which I am aware that were not overturned included things like negative recommendation from home bishop, candidate revealing discomfort with some of the university's policies or church doctrines. When I joined BYU, the hot button was whether we felt a need to discuss personal feelings about a female deity.