The M1903 Springfield is seen in the missions The Whore, where it is carried by a police officer. Then it's seen in You Lucky Bastard, where it is very helpful to use in order to kill the guards in the crane towers armed with the same weapon. It can also be found in Omerta, where it is used by an FBI agent trying to shoot Tommy from a small tower behind the hangars. If the player can shoot the agent he might drop it where Tommy can reach it, or else it falls behind a fence out of reach.
You can find more real world information about the US Rifle M1903 Springfield on its Wikipedia page.