The Department of Homeland Security (DHS) has a handy chart showing which nonimmigrant classifications allow study in the U.S.
One common violation of status our firm sees is that generally a B1/B2 visitor may only engage in study that is avocational or recreational in nature, and if a B1/B2 visitor applies for change to F-1 student status, the student may not begin attending school until the change of status is approved by DHS.