Factual error: There is, and never has been, for that matter, a Private First Class rank in the British army.
Deliberate mistake: The entire time they spend in the trench, before they attack, Quinn has a white sheepskin jacket. It rains throughout the film yet with all the mud about you would think in about a day that this jacket would turn dark brown. It is still fairly white even when Quinn dies.
Plot hole: Towards the end of the film when Private Shakespeare is talking to the wounded soldier, his coat and hair are suddenly dry, strange when the majority of the film is set outdoors in the rain.