Trench warfare was virtually universal during the Great War. In brief, innovations in machine guns and, in particular, artillery-- with no accompanying innovations in mechanized warfare, such as tanks, till the end of the war-- meant that any soldiers not burrowed into the ground would be blown to pieces before they could effectively engage the enemy.

So, when the Germans dug trenches on the far bank of the Marnes in the autumn of 1914 to fortify their positions, the British and French responded in kind. Thus began the famous stalemate on the Western Front, but this was a world war, with theaters in the Middle East, the Russia steppe, Africa, the Balkans, Gallipoli, and so on. And wherever there was fighting, there were trenches, for the reasons I mentioned in the first paragraph. Even in places where the fighting was far more open and fluid, such as the Middle East, which saw the use of cavalry and armored cars, soldiers would dig trenches around important strategic locations. If an attacking force was held up for any length of time, such as the Ottomans during the siege of Kut in 1915, they would be forced to dig trenches as well, and a situation not dissimilar from the stalemate on the Western front would play out.

I would say that it is possible that there were a small minority of places where trenches did not see widespread use, though sources do not say for certain. One of those places is German East Africa, where Paul von Lettow-Vorbeck waged a guerilla campaign against the Entente for much of the war. Trenches were largely unnecessary due to the nature of the fighting. Another place may be the Pacific Theater, where Japan (fighting on the side of the Entente) spent most of the war taking small islands from Germany, though there was a large battle at the very beginning of the war between Japan and Germany on the shores of China, IIRC.

Sources:

The Great War: A Combat History of the First World War by Peter Hart