On S&W revolvers, I recommend always using the serial number stamped on the outside bottom of the grip frame (will be under the bottom/base of the grips if you have the oversize target grips). Although modern S&Ws also have the serial number stamped inside the frame cutout, the older S&W models do not, and this can cause problems if an assembly or part number on an older model is recorded as the serial number on any important paperwork. The serial number is the ONLY number on the base of the grip frame, so it removes any possibility of mistaking one number for another, no matter how old the weapon is. There are exceptions to this rule for very old military-owned/purchased S&W revolvers, but for most common civilian models it should hold true.
I've seen gun dealers screw this up on several different types of handguns, so I know it happens regularly. I am also aware of at least one case where a police investigation was launched because a gun was believed to be stolen, because the owner of the stolen weapon had reported to police that a part number was the serial number. The wrong info was entered in a stolen weapon database (NICS), and eventually, the same thing happened again when a standard registration query was run. When a legitimate owner of a similar pistol (WWII Luger) tried to register a weapon that had been in his family for decades, he was hauled in and questioned for hours about a stolen weapon that he had nothing to do with; his only mistake was making the same mistake that the other gun owner had made, reporting a part number as a serial number on his registration paperwork.
"Placement is power" -- seen in an article by Stephen A. Camp
(RIP, Mr. Camp; you will be remembered, and missed)