Majority rule, as I've defined it, is not currently used in cities, largely for historical reasons. Until fairly recently, we haven't had the ballot-counting technology to make it viable. It is used by many professional societies, but those are smaller groups than cities.
However, alternative voting, which is very similar to majority rule, with ranked-choice ballots, has been used in many American cities. It's used in, for example, San Francisco, in Minnesota, and in my own town of Cambridge, Massachusetts.