Well, normally it was carried in a groove/hole in the stock under the barrel. Because it had to extend all the way down the barrel for loading ( and, by the way, cleaning- it was also used for that), about the only place to carry the ramrod was somewhere next to the barrel. There were tricks- like, sticking the ramrod into the ground between shots. But that wasn't in any drill. The drill would have the soldier replace the ramrod every time, because then it was not left forgotten sticking in the ground when it was time to charge, or wasn't left in the bore and fired at the enemy at the next volley. This is rather important, when you hear about tricks for fast loading ( the series Sharpe's Rifles made a big deal about one trick, for example): tricks were frowned upon for the rank and file. Because one of the difficult things about using muzzleloading weapons in battle was the number of steps in loading and firing: miss a step, and the gun would not shoot, or maybe not shoot next time. Run the ball down the bore without any powder ahead of it, and it would take some time to mount a screw on the ramrod to pull it back out. And, if the recruit was excited, it was possible he didn't notice if his musket hadn't fired, and he might ram another load on top the first. A calm recruit could drop the ramrod down the bore and tell, by how much was sticking out, whether there was a load in it, but an excited recruit might forget..and you couldn't tell how the gun was loaded by just glancing at it, so it was not unheard of for a dropped rifle to be found after the battle with several charges in it . Officers knew that if a recruit was thoroughly drilled on the same way of loading and firing, he would probably do it automatically in the heat of battle, without thinking. Which meant that he and his gun would continue to fire.

The famous story of many rifles being found on the Gettysburg battlefield, mis-loaded or loaded with two or more charges, seems to originate with Col. J.G. Benton's 1867 Course of Instruction and Gunnery for Cadets, pg 341. It's over on Google books.

To add to /u/bodark43's comment, I want to note that ramrods work best when they are more or less straight, particularly when dealing with rifled muskets (smoothbores are, as ever, wonderfully forgiving). Keeping the rammer tight underneath the barrel ensures that it is well-protected. Having it hang off the soldier's body is just asking for the ramrod to get lost, bent, tangled, and otherwise damaged.