A good example of what you're asking about may be the battle of Neuve Chapelle in March 1915.

Background

By December 1914, the Western Front had stabilized and both sides settled in for the winter, preparing defenses and planning for the spring offensives.

The British Army (now composed of the First and Second Armies) was on the left of the allied line. Sir John French, the commander of the British Army, tasked First Army, under the command of Sir Douglas Haig, to plan an offensive in the area opposite Neuve Chapelle (just on the German side of the line). The attack was originally supposed to be in coordination with a wider French attack, but by the end of February 1915, it had become clear that the French weren't going to contribute to the push. Sir John still wanted an attack, though, so Haig began planning.

The Battle

First Army's preparations were fairly extensive. The Royal Flying Corps photographed German positions, guns were moved into position, and ammunition was stockpiled.

In the morning of March 10, British guns pounded the German trenches for about 30 minutes (short by later 1916-17 standards). Much of the wire was cut, and the initial infantry advances captured most of their primary objectives. The village of Neuve Chapelle was captured as well.

The plan was for the assault units to capture the German positions and for the reserve formations to leapfrog the assault units and continue pressing on. Although initial attack captured its objectives, Sir Henry Rawlinson delayed moving the reserve elements forward until the afternoon, when he was certain the front was clear. Nothing really happened with the reserves until 3 pm, and problems with communication and relaying the orders meant that reserve units didn't actually start moving until 5 or 6 pm.

By then, German reserves had moved into place, and the opportunity to exploit the breakthrough had passed. The Germans dug french trenches on the night off 10/11 March, which helped seal the front. Fresh British attacks on 11 March lacked the same level of planning and coordination and thus faltered.

Analysis

Neuve Chapelle happened before the Germans instituted defense-in-depth, and so most of their forces were concentrated forward to defend the front-line trenches. The extent of aerial reconnaissance ensured that artillery hit these trenches and infantry knew what their objectives were and what they were dealing with on the other side of no man's land. This system confirmed that trench lines could be broken (at least before defense-in-depth came into play and changed the nature of the game a bit).

The breakthrough was not exploited for two reasons. First, Rawlinson was too cautious, and he failed to send the reserve IV Corps forward quickly enough. Second, when he finally delivered the order, it took several hours for the corps to actually start moving. It's not likely that exploiting the breakthrough would have had major operational consequences, however; the British Army was just too small.

The plan for the Somme in 1916 drew heavily on Neuve Chapelle, and the result of Neuve Chapelle taught Haig to have cavalry at the ready to exploit these breakthroughs. Replicating the success at the Somme was difficult, though, largely because the lengthy preliminary artillery bombardments alerted the Germans, who subsequently reinforced the front. A breakthrough at the Somme therefore remained allusive. Once defense-in-depth came into play, the prospects of breaking through the German lines were greatly reduced, and thus the concept of "bite and hold" became the doctrine of the day (see Vimy Ridge as a good example).

Sources

The best general studies are Sheffield, The Chief and Wilson and Prior, Command and Control on the Western Front. These books look the battle from the perspective of Haig and Rawlinson, respectively.