The military sniper is a tripod-mounted bolt-action rifle you only use once, in The Suburbs. A hunter uses it to pin Joel, Ellie, Henry and Sam down in the street; once Joel reaches the house and kills him, you man the rifle to cover the others.
Move between cars and houses to stay out of its sightline. Smoke bombs block the shooter's view and make the approach much safer.
The hunters' Humvee arrives during the fight and normal hits do little to it. Shoot the gunner as he gets ready to throw a Molotov: the bottle sets the vehicle alight and it crashes into a house.
Hatchet
The hatchet is a one-handed camp axe that kills any enemy except a bloater in one hit. It is the last bladed weapon you find, first turning up just after you leave Ish's abandoned settlement in the Pittsburgh sewers during The Suburbs.
Machete
The machete kills in one hit and swings faster than the hatchet, but breaks after three uses. The first one is in Ish's settlement in the Pittsburgh sewers during The Suburbs.
Shorty
The Shorty is a one-handed sawed-off shotgun that sits in your pistol slot. You find it in the Pittsburgh sewers during The Suburbs, on a bench to the left just after you set off the first sound trap.
2x4
The 2x4 is a plank of lumber and the most common melee weapon in the game, especially in Boston and Pittsburgh. It reaches farther than your fists, stuns a human with one hit and lasts four swings before it breaks.