In Pima County, AZ it is based on zoning. The zoning rules tell you how many horses (if any) you can own. It may differ between sides of the street.
The Pima County code can be found here: Municode
In my case, I need chapter 18 (zoning) and specifically 18.21 "CR-1 Zoning". It tells me "On lots of not less than thirty-six thousand square feet not more than one head of cattle, horses, sheep, goats, or other similar animals more than six months of age per ten thousand square feet of lot area, excluding swine..." - so 1 acre can have 4 horses.
I use this as an example because the courthouse wouldn't have a clue. The zoning people live in a different building.