Where is the serial number on 12 gauge American gun?

It may not have one. Assuming yours is a break-action shotgun, they were made by Crescent. Crescent made over 2 million shotguns under about 425 different trade names (American was the name used for guns made for Sears). Shotguns were not required by law to HAVE a serial number until 1968, and your gun was made well before the, HOWEVER- if there is a serial number, it is likely found on the portion of the frame that is covered by the breech end of the barrels. Remove the barrels, and look at the part of the frame that they concealed.