The Chief of Ordnance stated in 1864 that "the use of breech-loading arms in our service has, with few exceptions, been confined to mounted troops". The numbers show it. The Army bought 80,515 Sharps carbines, 94,196 Spencer carbines, 55,567 Burnside carbines, 30,062 Smith carbines and tens of thousands of other breech-loading ones for the cavalry, between early 1861 and middle of 1866. In the same period , for the infantry, they bought 12,471 Spencer rifles and 9,141 Sharps rifles and a few thousand other breech-loaders. But they bought around 1.5 million muzzle-loading rifled muskets. Add that to the many rifled muskets already in service at the start of 1861, and you can say the regular soldiers did not usually get them for the War.

What makes the question a trifle tricky to answer is that the individual States also sent militia units. Those could be armed differently. Soldiers could buy their own breech-loaders, and I don't know how many did. The 13th Pennsylvania Reserve Regiment carried their own Spencer or Sharps rifles. While the War Dept. was understandably quite reluctant to purchase Henry rifles, knowing they would then have to undertake supplying them with their special ammunition, some soldiers were happy to buy them and they could have quite an effect. At the Battle of Franklin, some of the 12th Kentucky and 65th Illinois were equipped with Henry rifles: equipped with 16-round magazines, they made Hood's frontal assaults even more futile. But, you could certainly say that these were exceptions to the general rule.

Fuller: The Breechloader in the Service