Hi Tom! Most 15-1/2" Regina music boxes have the serial number on 
the bedplate right under the embossed word "REGINA". If yours
doesn't have the embossed name it is probably an early Regina with 
Polyphon movement.

Polyphon was the parent company of Regina and, in the beginning of
production, Regina used Polyphon movements until they could produce
their own. Once they were established with their own foundry and
machinery, Regina then had their name on almost all their 15-1/2"
machines.

Polyphon often put a serial number on the turntable that is on the
center spindle. These were usually four digit numbers and dated from
1894-1895.

Unfortunately, the coin mechanisms are usually missing in total or in
part. There were about 4 different configurations they used for this
mechanism, so there is no one particular design. Usually the only way
to find a mechanism intact is to find a junker donor box.

Enjoy!

Nancy Fratti - Nancy Fratti Music Boxes
Canastota, New York
http://www.nancyfrattimusicboxes.com/