Language, as a rule, has few, if any, clear cut boundaries. This is very true in America where people tend to move frequently. I would say that all three (raise, bring up and rear) are fairly common throughout the country. I was taught that you raise an animal and rear a child. I believe that, "to rear a child", is used more in the Mid-Atlantic area - south of New York. In the mountainous region one can still hear archaic English terms, some going back to the 15th Century or earlier.