Omega serial numbers are either laser etched in tiny digits on the case back or for older watches it may be on the inside of the case. It will be a number (no letters) between 5 and 9 digits long. The number containing dots, spaces and possibly letters is not the serial number, that is the model or case reference. These can be decoded using the Omega model decoder.
You can also look up the date from the Omega movement/caliber number etched directly on the movement itself using the Omega movement database.
When combined with the information from these tools you can help to validate an Omega's authenticity or to identify a fake Omega. This tool can date vintage Omegas from 1895 to 2008. Speedmaster serial numbers did not run inline with non-speedmaster serial numbers so please select which you have, if you have a speedmaster it will be clearly marked on the case back.