Originally Posted by bhomburg

Neither SQ nor any other airline has security cameras on aircraft. There is no footage, and if the crew didn't see anything you're pretty much SOL.

You'll need to file a police report, and if you are very lucky the police will check with the crew (or, armed with the police report, email SQ in Seoul using their online form available via their website.
Practically, the only chance would be if a crew member saw something like the passenger in 55D getting up in the middle of the night and opening the overhead bin across the aisle and remembering it.
Chances of recovery are virtually zero.