The great stereotype for the hopeless First World War battle fought in the mud, the filth and the rain is Passchendaele (formally known as the Third Battle of Ypres), fought in Flanders from mid- to late-1917. The ground around Ypres is low-lying and the autumn weather wet. Prior to the war, the whole region had been well-drained and cultivated, but three years of fighting had utterly destroyed the ground and the drainage. This was compounded by Passchendaele being the first major British offensive effort in the Ypres Salient since 1914, and it was preceded by a long, heavy bombardment which only further ruined the drainage. The battle began on July 31st, the rains began in the afternoon after three thousands yards of ground had been taken, and by August 2nd had literally forced a halt in all operations. Men had to be sent forward on duckboards over the seas of mud. Muck had to be hosed off shells before they could be fired. The Gheluvelt plateau, which had been a first-day objective, was only taken after the weather briefly cleared on October 4th, but then the rains resumed. The battle would continue until November 10th, when the British finally gained Passchendaele Ridge as a secure line of defence for the winter.
Sources:
Gordon Corrigan, Mud, Blood and Poppycock
Saul David, 100 Days to Victory
John Keegan, The First World War
Gary Sheffield, The Chief: Douglas Haig and the British Army
Simkins, Jukes and Hickey, The First World War: The War to End All Wars
Hew Strachan, The First World War
To add to /u/RobWithOneB's answer, John Hussey did an essay on the weather during the 3rd Battle of Ypres, in which he found that based on meteorological records from nearby Lille, western Flanders at least had been tending towards dry summers and autumns, with the weather breaking in November, for the past few decades. The weather at 'Passchendaele' was thus extremely unseasonable, although it did dry up at the end of August with rain resuming in early October. There were rainless spells in August and July, but the cloud cover prevented the rain from evaporating.
The Battle of the Somme saw dry weather in July and September, with spells of inclement weather in August, and then rain and snow in October and November. In both Third Ypres and the Somme, dust was a huge problem for aircraft and transport vehicle engines.
In WWII, the soldiers of the Allied Expeditionary Force had to endure the wettest, coldest autumn and winter on record in Northwestern Europe, which had unfortunate implications for air support. In 1945, American, British and Canadian troops had to fight a veritable amphibious campaign in the northern Rhineland, as the Germans had broken the Roer River dams, and flooded the valley.