I haven't been able to find one on mine.The Savage model 342 was produced from 1950-1955.During this time frame serial numbers were not required on firearms.The passage of the gun control act of 1968 mandated that all firearms made in the U.S. and abroad have serial numbers,and this was started in the year 1969.I believe that your Savage model 342 were not given serial numbers at the time of production.