First thing I want to address is the fact that this topic usually comes up when discussing what to do with a weapon while we are moving. Navy SEALs for instance have religiously used the tight muzzle up position while on the move and at the ready. Their reason for this will be discussed, but I just wanted to note their method. The rest of the military seems to stick to a downward muzzle position for everything.
The idea of picking one of these positions, when looking from an outside perspective, is to have a position that will result in safe handling, while allowing you to quickly employ the weapon upon your target. In my experience, both positions allow this, but in different ways. Each one has an advantage and disadvantage that need to be realized.
The idea that we are over-concerned about the safety of carrying our weapon in a certain position is pretty ridiculous. The source of safety with any weapon is where your trigger finger is, and where your brain is at the time.
The other thing I want to touch on is the aspect of speed. Being that the two positions have their differences and one may be slower than the other in a matter of micro-seconds, I feel that the complaint of time into action is quite petty. But in some cases such as room clearing, time on target is a big deal. But then again, a practiced expert or professional will see very little difference in the time it takes to put rounds on target from their preferred position.
The muzzle up method is very popular, to the point of a religion, with the Navy SEALs. The problem I am noticing lately is that people are advocating for the muzzle up position without understanding why the SEALs even use it. Well, to be brief, they mainly use it to be prepared for hand to hand contact with the enemy. As a direct action force, it is not uncommon for them to have to go in to capture people and take them prisoner. When they enter a room, the rifle is supposed to be brought downwards on target, but with the rifle ready in the up position while entering in order to help facilitate strikes on the target if the need arises.
When muzzle up is being employed by the SEALs in movement, you will notice that it is straight up and locked against the body. this allows the shooter to move quickly and limit the possibility of muzzle sweeping or snagging your muzzle. I also would say that this position is alot better if you were to take a fall. But I would say that in movement, the muzzle up position feels very tight and very good.
When going from moving to getting on target, you need to pull the rifle back into your shoulder and then down onto the target. This can take time to practice so you don't risk missing your shoulder. For the most part, though, I would say that the only gun I would say needs a full support of the shoulder is the shotgun.
Other than a few examples of what the muzzle up position is capable of I feel it has a definite place and should not be ignored. I will go over how I suggest using it in a little bit.
When you look across the military branches, you will notice a majority of the services employing the muzzle down position. I have found that this is a very good position to be ready to engage targets efficiently. You can rest the rifle downward and still have the butt of the rifle in your shoulder, ready to go.
When using the muzzle down method during movement, I have found that it does cause flagging at times without people even realizing it.
Just as an example, I watched an ANA soldier accidentally shoot himself in the foot while running in the middle of a firefight. He was just using what he was taught, but failed to put his gun on safe and keep his finger off the trigger. When the rifle goes off, some say this method is safe because the round will stop in the dirt. This is not the case all the time. Hit a rock or hit asphalt, and you instantly have fragmentation and ricochet. Beside the risk of a round going off, I found that the muzzle down method can cause snagging in many cases. I have found this to be the case many times when doing training in many different places.
Overall, I don't think that any one method here is superior to the other, except in specific application. I found myself not always using the muzzle down method when I was actually fighting. I ended up fighting from the heart and using whatever I needed to use to be faster, more efficient, and more deadly. I would encourage you to also consider this. It is about having the skills to survive, not to look like a Hollywood Spec-Ops poster boy. Use what works for the job, not what looks the coolest while doing the job. Basically, don't use a hammer in a screw, and don't screw a nail in.