The United States had two primary types of sniper rifles during World War One, although both were based on the M1903 Springfield rifle.
The most common optic used was the Warner & Swasey "Telescopic Musket Sight", a rather clumsy prismatic optic mounted on the left side of the rifle, on a detachable rail. The model 1908 W&S offered 6 power magnification, which was reduced to 5.2x in the 1913 model in an effort to increase field of view. These optics were also used on the M1909 Benet-Mercie light machine gun.
The second type is the Winchester A5 scope, an excellent commercial scope available at the time. Although usually associated with the US Marine Corps, several hundred of these were also issued by the Army. The A5 was a much more tradition type of optic, mounted centrally above the bore and preferred by competitive marksmen.
The third rifle we are looking at in this video is a very interesting example of a competitive rifle from the pre-WWI period. It is a 1903 Springfield fitted with a commercial A5 scope and Mann bases. This is the sort of rifle that would have been used by the career military shooters for competition, and would likely have accompanied many such men overseas in the American Expeditionary Force. Woe to the German who found himself in the sights of such a man with a rifle like this!
WWI Periscope Trench Mauser: https://www.full30.com/video/5e70aefd851ade5cdc95f9b1f5d533ff
WWI German G98 Sniper: https://www.full30.com/video/61de6f7f6d2eb686c390d16e2bd9ac8f
US WWII Springfield Snipers: https://www.full30.com/video/f7b62b7c1e1c7d8b57bd64e6f78b59c5