I don't know all the rules, but I can share my experience. Last year, when I went to the courthouse in Sundance to do my car titles and registrations, I also registered to vote. It is pretty convenient that the county clerk, assessor, and treasurer are all in the same side of the courthouse and usually there is no waiting.
I went to the county clerk's window and told the lady that I wanted to register to vote. It seemed to me that she gave me a look like "Oh, really. I am not sure that we want any more voters here." Or maybe there are just not many people registering to vote and she was temporarily taken by the novelty of my request. She opened up a drawer under the counter and pulled out a form asking for my name, address, etc. for me to fill out. I do not believe that there is any particular minimum residency requirement, but the lady was serious about checking my Wyoming driver's license and my information on the registration card. Since I had been previously registered in Nevada, she also wanted me to fill out another card renouncing my old voter registration. Then she gave me my voter registration card to bring to the next polls.
The whole experience made me think that the county people are very serious about voting, and as far as reasonably practical, to prevent non-residents from voting. A few months later, I even got a call from the county clerk's office. It seems that I left my party affiliation blank, and she wanted to fill it in so that my voter registration was complete. Of course, there is the chance that she was also confirming my residence too...