Where do you find the serial number on a pair of Beats Wireless Tour 3?

Try this:

https://support.apple.com/en-us/HT204196

Do you mean it's not listed on the product or the Beats site?

It should be listed both on the product itself and in the Beats app for iOS.

Try this link instead.

https://www.imore.com/how-register-your-powerbeats-pro

The link is for Powerbeats Pro, but the process should be the same. It should be listed in your phone settings as well.