I have 2 styles, a couple of years old, so I can't speak to the Muse, if that's the style that is your concern, but one of mine has a serial number on the back of a leather tab inset into the back (a logo tag or signature tag--I'm not sure what the technical term is for it) and the other has a serial number printed on the lining inside the flat interior pocket. So, yes--both of mine do have serial numbers inside of the bag. But, I don't know if older ones do or if this year's handbags do.
I just bought a Mombasa through YSL. The back of the leather tag that has "Yves Saint Laurent Rive Gauche" in silver lettering has a serial number and says "Made in Italy"--those portions are stamped into the leather.