The Highland Rim Historical Society believes it is the plural: Cold Springs School. This is based on documents we have seen. For example, see the Thomas Baskerville Water Agreement, the Howard McDowell Certificate and the 1923 Portland Herald article on this page. There are other documents, however, which refer to just one spring. See, for example, the 1857 deed on this page. When the Tennessee Historical Commission erected a historical marker in Portland, they chose to use Cold Spring School. We don't know why they chose the singular.
The location and number of springs in this area change from time to time. At one time, there may have been multiple springs. It's really impossible to say.