Knowing their usage was limited to US forces, how often were troops equipped with one, or had one for use in their trenches?
By trench gun, I assume you mean the Winchester model 1897. Known for being unpopular due to its ability to increase suffering. They were initially brought in as not only a weapon, but as a way to shoot grenades out of the air, as the US army used skeet-shooters to do so. The gun itself used paper cartridges and with the weather of France not known for being clear, this diminished the use by a lot. Failure to fires and loading problems were common, couple that with the intense hate from the Germans (going so far as to release a statement that they would execute any soldier found with the gun) the gun was few and far between on the Western front and isn't as synonymous with the war as you would think. Mostly they were used in the rear by the MP's, not only for guarding POW's but also for keeping rowdy Doughboy's from acting up when they had too much French wine.