Finding serial number for a device - https://discussions.apple.com/message/18246553 - from box or backup file, not from Apple.
http://support.apple.com/kb/HT2526 - If you registered a device you can use My Support Profile to find a list of serial numbers that have been purchased or registered with your Apple ID.
My Support Profile - https://supportprofile.apple.com/MySupportProfile.do
Maybe a way to find serial number after a device is no longer in your possession - https://discussions.apple.com/message/19524469 - "The easiest way to find your serial number is this Open Itunes...click help... run diagnostics.....(tick or untick as necessary) so that you just have ticked the last two, device connectivity and device sync. run these tests. it will say no ipod/ipad found but will give results of test....scroll down to bottom of results... here you will find the serial number of your last attached devices."
iPhone: Locating the serial number, IMEI, ICCID, or MEID - http://support.apple.com/kb/HT1267 - from phone itself, a computer to which it has been backed up, or packaging.