The American Army has always had an on-again, off-again love affair with the rifleman. The twelve company strong First Pennsylvania rifle regiment was raised by Congress and disbanded after their one year term of service expired. Washington then had Daniel Morgan raised Morgan's Rifle Corps which had men detailed from their parent regiment. It was an ad-hoc organization and except from two companies was disbanded when the men were needed to fill out the depleted ranks of their parent regiment. Another ad-hoc battalion (Major Parr) was raised and served in New York. After the Revolutionary War, there were no rifle regiments in the American Army until a little before the War of 1812 when the First Rifle Regiment was raised. They were supposed to be dressed in green coats and equipped with the Model 1803 rifles. Trouble is, there weren't enough green coats (or they wore out) to issue and because of the shortage of rifles, some men received muskets. Three other rifle regiments were raised during the war, but the supply problem persisted and by war's end, all the regiments, save for the 1st, was disbanded. The 1st was disbanded later and the next rifle regiments was really a volunteer unit (Jeff Davis regiment of Mississippi riflemen) during the Mexican-American War. A mounted rifle armed infantry unit was raised, but they were soon converted to cavalry. In the Civil War, we had no sharp shooter unit and the new ones were volunteer organizations (Berdan, Birge, Andrew Sharp Shooters, Brady's Michigan Sharp Shooters). We all know what happened to the volunteer organizations. Some were disbanded during the war (enlistment expired) and none survived in the post-Civil War frontier army. Thus, at least one ad-hoc unit was raised in the Plains War and during the Spanish American War men were again detailed to counter the enemy's sharp shooters. World War I & II, Korea, Viet Nam all followed the pattern established during the American Revolution.
A bit of digression. It was mentioned that most sharp shooters were not snipers. This is correct as most were skirmishers. In fact, the term sharp shooter was very loosely used (see Gettysburg #39) by the Civil War soldier. There's a lengthy discussion about the four classes of sharpshooters (from the blind squirrel to the Olympic quality champion) in the book. One thing to remember is that there are no hard and fast rules and there are always exceptions. For instance, I found men who, if they had the money, bought telescope target rifles probably so they could stay out of the infantry charges.