This is an interesting question since most insignificant passages are overlooked by the reader. The author's purpose for mentioning them is vastly more difficult to determine.
1. "A negro teamster who had been dancing upon a cracker box with the hilarious encouragement of twoscore soldiers was deserted. He sat mournfully down." The event, following the soldier's new gossip that the army may be moving, is insignificant except for the fact that what had been an entertaining moment just seconds before has suddenly been forgotten. Crane shows this event to illustrate how boring camp life has become. (Chapter 1)
2. "As the horseman wheeled his animal and galloped away he turned to shout over his shoulder, 'Don't forget that box of cigars!' The colonel mumbled in reply. The youth wondered what a box of cigars had to do with war." The event is totally insignificant, and we hear no more of the horseman or the colonel or the cigars. Crane probably added this scene to project the enormous senselessness of war with yet another senseless act. (Chapter 2)
3. "A rather fat soldier attempted to pilfer a horse from a dooryard. He planned to load his knapsack upon it. He was escaping with his prize when a young girl rushed from the house and grabbed the animal's mane. There followed a wrangle." The men later called out to the girl to "hit him with a stick." The scene is certainly a comic one (and in the film version as well), and this is probably Crane's purpose for including it. The scene is filled with irony: The men are happy to be moving (they assume that being shot at is better than the boring camp life), and they think it funny that a Union soldier cannot commandeer a horse from a young lady. The Rebel troops they face later will prove to be a tougher match. (Chapter 2)