The most "common" sniper rifle is the Remington Model 700 PS (police sniper). There is also a variation of the 700 made for Varminting which is also used by many police forces.

The calibers they come in are .308 and 7 mm mag.

Scoped "typically" with a 3-9 Leupold tactical scope.

For cheaper variations look at FN and Savage. Sako also makes a really nice sniper gun (more pricey than all of those mentioned). I saw a used 700 PS with leupold scope for 800 bucks, almost purchased it, but really have no use for such a heavy gun.