To give a more dramatic illustration of the capabilities of what at least some professional armies in Europe could do in the time period, I commend to the gentlemen of the thread the following. The troops engaged in this behaviour are not specially trained compared to the rest of the army.
1) Inkerman.
Some line regiments shake out into skirmish order and outshoot Russian artillery batteries at several hundred yards - that is, they're able to engage artillery at the range of artillery. It shouldn't be hard to see how useful this would have been in the Civil War - imagine if the Confederate troops going up against the Union on day three of Gettysburg could have stopped five hundred yards away and sniped out the Union artillery!
(from With the Connaught Rangers)
2) Crimea more generally. This time from the History of the Rifle Brigade, which was a regiment ranked 28th in the army in accuracy of musketry.
On October 13 a man of the 2nd Battalion, Herbert, made a most remarkable shot. He was on outlying picquet, and observing a Russian officer on a white horse he took a shot at him, fixing the sight of his rifle at its extreme range. The officer fell, while the horse moved on. The distance at which he shot him has been variously estimated from 1300 yards downwards ; the man himself told me that he thought the Russian whom he shot was about 1000 yards from him.
On October 14 Fyers was with his company in the five-gun battery when he observed a column of Russian infantry advancing. Taking a rifle from one of the men, he put the sight at what he considered their distance, and fired, carefully watching the effect of the shot. When he perceived that it struck the ground a little in front of the column, he ordered his men to fix their sights for 750 yards, and to stand up on the parapet and ' give it them.'
As another instance of their excellence in shooting, I may add that Atherley, in the course of this day's fight, asked one of his men, named Robertson, how far he estimated the distance of the brick-kilns to be. The Rifleman replied that he did not know ; but calling Atherley's attention to a man standing on the top of the kiln, he put up his sight for 600 yards, fired, and the man fell. His body was examined the next day by Atherley, and the ball was found to have hit him in the stomach.
Shooting at battle range:
Some time after, a large body of cavalry appeared in their rear (the proper rear of the column). These were at first taken for Hodson's Horse ; but infantry soon appeared, and it was ascertained that they were enemies. Two companies of Riflemen moved down into a hollow which afforded good cover; and as the cavalry passed, gave them a volley at about 500 yards. This the infantry returned with a straggling fire and then turned and fled.
Firing at unusually small targets:
On one occasion about this time, when a party of the Regiment had been pushed forward, four Riflemen crept up to within 500 yards of the place and fired into the windows of the grand barracks of Sebastopol.
The attack was confided to the Riflemen. Brigadier Horsford's orders to Colonel Hill were to advance to within 400 yards of the fort: then to open fire on the embrasures. Mortars and heavy guns were ordered to the front, and cavalry to the flanks. This took some time. Then the Battalion advanced to the front face; two companies skirmishing ; two supporting them ; two moved to the left; the remainder in support. A sharp fire was opened, and was returned for some time by a fire of grape from the fort. The Riflemen continued their fire for about two hours, picking off the gunners at the embrasures.
Suppressing artillery:
On the 19th a man of the Regiment was seen to pick off eight men from a Russian battery.
The enemy brought forward a field battery of six guns and opened fire on the line. This fire became very troublesome on the right flank, and Lieutenant Godfrey with a few men was sent to try to silence these guns. This they did most effectually in a very short time. The task was difficult, for the ground afforded no cover ; the utmost shelter they could get being some slight undulation in the surface. However the Riflemen lay down on their stomachs and picked off the gunners whenever they attempted to handle their guns ; and in about twenty minutes forced the Russian guns to retire.
These Riflemen getting cover in the brushwood on the left of the Barrier picked off the gunners of the Russian battery on the Shell hill. [c.700 yards]
General Windham ordered him to line the bank of the canal. Three guns were brought to bear on these Riflemen, and several round shot came amongst them, but without doing any hurt. Atherley made his men take shelter along the bank ; and selecting two whom he knew to be excellent shots, he told them to pick off the gunners of these guns, which were annoying the troops from the bridges over the canal ; and he desired some of their comrades to load for them, and to hand them up rifles as fast as they could. Thus aided, these Riflemen, creeping up near the bridges, picked off the gunners, and effectually silenced the guns.
3) Accuracy stats.
It is generally considered that at Inkerman and the Alma the typical battlefield accuracy of British troops was on the order of one hit for every sixteen to eighteen shots fired. By contrast, at Gettysburg Union accuracy (with a much shorter general engagement range) was on the order of one hit for every one hundred rounds fired.
The British in this are exceptional - their focus on rifle training was the most obsessive in Europe. But other armies were better at other things - the French conducted balls-out uphill bayonet charges at Solferino, for example, and the Prussian artillery was an absolute terror.