There aren't many reasons why you would want to remove a serial number from a weapon. The ones that do exist are usually related to a crime that was committed with that weapon. The idea behind it is to make it difficult to trace the gun to its owner or the person to whom the weapon is registered. The fact is that law enforcement does not need that number to trace a weapon that has been used in a crime. In addition, even if the number has been etched off completely, there are ways to recover the number.
Unload the weapon prior to attempting any of these methods. Remove bullets from the revolver and set them aside. For semi-automatic pistols, remove the clip by pressing the button or lever just to the side of the clip and pull it out. Pull back on the slide and make sure there isn't a round in the chamber.
Inspect the weapon. Many manufacturers place the serial number in multiple locations on the same weapon. For semi-automatic weapons, the number is often placed on the barrel and slide inside the gun. For revolvers, the number can be on the barrel or metal piece under the grip. It's possible that the person attempting to remove the number may miss one of these locations.
Polish the surface where the serial number was located. Magnetize the weapon with a strong permanent magnet. This process causes magnetic waves in the area where the number has been filed off. Spray a suspension of magnetic particles in oil on the surface of the weapon. The oil will cause the smallest of particles to settle into the grooves where the number was located, revealing the number. These first two methods are preferred to the remaining two because these do not permanently damage the weapon.
Brush an etching solution onto the area where the serial number was located. This chemical eats through the scarred area and reveals the serial number. This method will alter the weapon, and the chemical could destroy evidence relating to this gun.
Set the weapon in a plexiglass box filled with distilled water and apply ultrasonic vibrations to it. The vibrations produce bubbles that result in a process called cavitation, which eats away at the area that has been filed off and will settle into the original grooves of the serial number.