I understand the main S/N is on the yoke just above the model number.
This is not always the case. The number on the frame inside the yoke cut is sometimes
the serial number, but it's often an assembly number, which was used during production but became meaningless after the gun left the factory.
The serial number on a S&W swing-out cylinder revolver is almost always found on the butt, i.e. the bottom of the grip frame. On some pre-WWII models that shipped with factory "extension stocks" or oversized grips that cover the butt, the serial number was stamped on the frontstrap above the butt.
The number stamped in one of these two places is THE serial number. Differing numbers in other places should be disregarded.
Other minor notes:
- A S&W lacking serial numbers in both the butt and frontstrap locations has probably had the serial number removed in violation of federal law. However, prior to the passage of the Gun Control Act, S&W built a few "club guns" (for giveaways at special events) with serial numbers consisting of one or more zeroes, and there are a few non-serialized "lunchbox guns" floating around that were snuck out of the factory in pieces by employees, then assembled at home (kinda like the song "One Piece At A Time" by Johnny Cash). Such guns can be valuable in collectors' circles but should be approached with extreme caution by novices.
- S&W did not place the serial on the frontstrap of postwar guns with oversize grips. On such guns, you may have to remove the grips to see the number. [EDIT: Highpower3006 beat me to it re: this one.]
- On guns built prior to ~1950, the serial number was repeated on the cylinder face and the underside of the barrel above the ejector rod. On some guns built prior to ~1920, the number is also found on the back of the hammer below the cocking spur. However, the numbers in these places should only be used to verify whether the gun is a "numbers matching" example with all-original parts. These numbers should NEVER be used for formal recording of "the" serial number, as these parts may have been replaced with parts from another gun.