I'm not entirely sure but there are probably at least 3 or 4 nicknames going around for my mayors right now in each of my two towns. I always agree to their request regardless of what they want to call me because I know it won't last.
In Sakura, I know that Mint and a few others have started calling me 'bucko', which I don't really like. Hans calls me 'chickpea', which is kind of cute. And I don't remember the others.
In Nocturne, O'Hare has started the trend of calling me 'Super A', which is kind of meh in my opinion. But Sylvia calls me 'dearie', and I kind of like that. Those are the only two I remember but I know there are a few more nicknames floating around.
I generally don't tend to like the nicknames that the villagers come up with, so I either just have them keep using the actual names of the player character in question, or a shorthand version thereof.
i got twiggy from someone on a forum and she came with the nickname "kinggggg" so i changed everyone else's nickname to me to kinggggg or king lol. besides that, some of them call me major r or just r
Only one at the moment, which is Young L. So far, half of my villagers calling me like that. It's not my favorite nickname, but also not the worst, so if everyone wants to call me like that, then I'm fine with it.