Recently, I've stopped Caroline, Marina, and Chief from moving. I never want Caroline to move, I like Marina as well, and Chief hasn't been in my town for very long (less than a month or so).
The last villager I had ask to leave was either Lolly or Deli. Its been a while before anyone has asked though because Ankha has been moving in the last few days.
None of my villagers have asked to move yet, but if I had to stop one, it'd be Punchy and maybe Nana, the problem with Nana is she moved near the front of my house, so I may or may not let her stay. :3
Ruby makes me feel sick. She sorta creeps me out, lol, so when she asks to move, I'll happily say goodbye.