If you cabinet doesn't have any hackery going on.....the 12 volt line is tapped off the transformer. This is unregulated 12 volts AC.
120 volts goes into the transformer from the line cord.....a 12 volt output winding leaves the transformer......one wire goes through a fuse.....and both wires end up at a connector....and travel on to the bulb sockets....where they're daisy-chained to each socket.
Check your fuse.....follow the AC with a meter.
Edward