S&W generally didn't use zero at the beginning. Serial 123 was not 0123 but just 123.

Still sounds like a M&P Model. Serial 508XXX would be from the late 1920s. The S prefix was added at the end of WWII but the numbers would be in the 900,000 range, not 500,000.

It's very possible the gun has been refinished (Are the hammer and trigger nickel plated? if so, sure sign of refinish as S&W never plated these) and the serial stamping partially buffed away.