They did. They did a lot in fact. There were regular "saps", which were small dead end trenches that dug out into no man's land. They were designed as listening posts to gather intelligence from the enemy, be early warning for attacks, or as jumping off points for your own assaults. They also did what was known as mining, where they dug tunnels under no man's land. They would either lay explosives under the enemy position. Miners were often commercial miners in civilian life, and the idea (for the British, at least) actually came from these soldiers who exploited their own expertise. Miners had to be extremely careful, as the trenches were close enough that the enemy could sometimes hear the miners and detonate explosives to destroy enemy tunnels. Opposing tunnels even met underground sometimes, leading to terrifying close quarter battles in the dark.

Perhaps the most famous mining action in WWI was the Battle for Messines Ridge, the opening move of Passchendaele in 1917. The British spent years digging under the German position and prepared a number of mines that were detonated at the beginning of the assault. It was said that the sound of the explosion could be heard in London. The general in charge of the battle told his soldiers:

"Gentlemen, we may not make history tomorrow, but we shall certainly change the geography."

Ultimately, digging towards the enemy was a part of trench warfare. It did not prove decisive at all in the war as a whole because the frontline was a lot more than a single line of trench; it was a whole network of trench lines, with significant depth to prevent an attacker from breaking through with one push. You might be able to break the first line of trenches with mining, but you would expect to be counterattacked by the second line of trenches almost immediately, which meant that your hard work was not useful unless followed up by an combined arms assault.