When SEAL Team Six operators run with a rifle, they're often running "muzzle up." In contrast, Delta operators usually run "muzzle down." Which is better? In the next three videos, Kyle Defoor (SEAL) and Paul Howe and Kyle Lamb (Delta) discuss their views:
Depending on the situation, I've used muzzle up and muzzle down, and environmental awareness is critical: don't aim at anything you don't wish to destroy/kill. If I'm loading onto a boat, I don't want to put a hole in the deck. Likewise, if I'm loading onto a helo, I don't want to put a hole in a main rotor blade. Also, I wouldn't want to aim my muzzle up at my buddies on the floor above me. On the other hand, trekking through deep water or mud, it can be practical to carry the weapon up out of the muck,
Both muzzle up and muzzle down can be used to run fast as is seen with Defoor's example above and Pat McNamara's (Delta) example below:
As shown in the photo below, SEALs have been going "muzzle up" since Vietnam:
But they also used muzzle down.
Another important factor to consider is training: follow the rules of the instructor/range. If the instructor is a SEAL, muzzle up will probably be allowed more, but if the instructor is a Delta operator, he'll likely expect much of your movement to be with the muzzle down. When in Rome, don't piss off your instructor. In summary, situational awareness and safety are key in deciding whether to go muzzle up or muzzle down.