There are several ways you can find the serial number (S/N) for your iPhone. You can also find the International Mobile Equipment Identity (IMEI) number, and identifier or ICCID if you need because them stored in the same location as the serial numbers.

If your iPhone is turned on choose Settings>General>About and scroll down to find the serial number and all other information about your phone.

If your iPhone is turned off, check the barcode on the packaging your iPhone came in to get the serial number and other identifying information.

If you don’t have the packaging for your iPhone and it won’t turn on, open iTunes from your computer. If you have a MAC hold the Control key on the menu bar at About iTunes. If you have a windows computer choose Help>About iTunes.

Resource: iPhone FAQ