What is the safest way to carry a long gun?

Depends whether it is a rifle or shotgun. For the latter, on the shoulder muzzle pointing skywards (which I prefer) or under the arm, muzzle point away from you and at the ground. Rifle - under the arm, muzzle pointing away from you at roughly 45 degrees to the ground, or sling over shoulder - in which case the weapon should be unloaded. If you carry the gun loaded (for hunting) always unload the weapon and place on ground before navigating obstacles.