Presuming we is the US, after the Civil War, when they started converting surplus Model 1861/1863 Springfields into trapdoor (eg breech-loading) rifles. Muzzle loaders would continue in service for a while (up until 1872) due to slow procurement of weapons. You can still find parts of the world where people use them, though. They've been handed down for generations and are heirlooms more fit for a museum than a war, but they're out there.
By the 1870/80's just about all major powers would have at least been in the process of re-equipping their armies.
There are single shot rifles still in use for Drill Teams, USMC shooting teams and a variety of bolt action sniper rifles. Muzzle Loaders slowly went away after breech loaders slowly filled in. The first full auto rifle was the BAR and until the M14...but I know that many were altered so they could not select full auto...that big round created a problem in full tilt boogy...hard enough holding down 10 rounds full auto in a M16. We're back to single shot is still in use.
The civil war was the end of muskets. However Henry rifles (lever action) were used throughout the entire civil war.
Muzzle loaders were replaced in the U.S. military with the adoption of the Model 1869 single-shot Trapdoor Springfield Rifle. The last model of the Trapdoor Springfield saw use in Cuba and the Philippines during the Spanish-American War. That was the last use of a single-shot rifle by the U.S. military.
