Clarinet manufacturers engrave serial numbers onto the wood or plastic of the instrument. Typically, these serial numbers are on the bottom of the upper joint or on the top of the lower joint, near the thumb-rest.
Most clarinet makers have a list of serial numbers and manufacturing years. Thesax.info provides a comprehensive list. Additionally, Woodwind.org has links to serial number lists for most brands. Knowing the serial number and year is helpful when appraising or selling an instrument. It is good practice to keep a record of instrument serial numbers to use in the case of instrument theft. If a student is using the instrument in school, knowing the serial number is important in avoiding an instrument mix-up with classmates.