There were a couple of methods. The first and large scale way to take an enemy trench line and fortifications was to have the offensive start off with a creeping barrage (also known as a walking barrage). Meaning the unit that was about to cross no-mans-land would get a shield of artillery fire, that would walk them towards the enemy trench. The artillery would first fire an opening barrage for a while on the enemy trenches, to hopefully kill the soldiers, or at least daze them. Then a second barrage would begin; the "Creeping Barrage". This is when the whistle and signal to go over the top would be sounded. The advancing troops would climb over the trench wall in to no-mans-land. They would get a screen / shield of artillery that would be highly coordinated beforehand and during the advance. This artillery would be shot right in front of the advancing troops. And as the troops advanced, the artillery barrage would advance with them, closer and closer to the enemy trenches. When eventually the artillery would start raining down on the enemy trenches again, killing, maiming, or dazing the enemy occupants. Then the offensive soldiers would jump in to the trench, usually with bayonets fixed to their rifles, spades, shovels, trench knives, pistols, and shotguns ready to go. This was close quarters, oftentimes hand to hand combat. And after the first wave of soldiers (or in the German's case, the Stormtroopers) had killed most of the enemies in the trench, a second supporting wave would come in, and help mop up, and fan out clearing the trenches. And that is one of the ways trenches were cleared. Of course Creeping Barrages had to be extremely accurate and precise, or else they would accidentally kill their own soldiers with friendly fire. So it was very dangerous. This was an allied method that was often employed, but both sides used this way to clear trenches.

The second method, and this was a largely German method, was through sneaky techniques. They would maybe drop smoke, or wait until the middle of the night, or whatever. Usually have little to no artillery support. They would send stormtroopers and engineers forward through no-mans-land. These soldiers would creep and sneak their way up to the enemy trench line. Then they would throw a ton of grenades, and satchel explosives in to the trenches. Taking the enemy by surprise. Then the stormtroopers would jump in, and with pistols, knives, shovels etc... would fight hand to hand and attempt to take the trench that way. Oftentimes the Engineers would use flamethrowers to burn out Allied troops. The first strikes were Knocking out Machine Gun and communication positions first. Then a second wave of supporting regular infantry troops would flood in, and clear the trenches.

Also gas was used similarly. But that's how many trenches were taken, from both sides, during World War I. Once tanks were really used, that also added another element in to the battle - with the biggest element being fear. Nothing like seeing a giant armored behemoth barreling towards you to make you turn tail and run.