It depends what kind of job you're applying for. Some have a more in depth screening process than others.
I personally hire people with some regularity. I have yet to ever call a reference. As long as they provide me with transcripts or a DD-214 with a resume that does not look like it was made by a child, and they seem personable enough, I will hire them. But the positions I'm hiring for are very entry level. For something requiring more demonstrable experience or something with security concerns, I'd expect them to call at least your most recent employer or ask for a professional reference.
I'm actually emailing an old supervisor right now to request that he email my background investigator at the agency I'm hoping to be hired on at. He doesn't know about my transition, so I've got to remind him who I was, explain who I currently am, and get him to agree to talk to someone else on my behalf. I'll let you know how it goes