I have been trying to understand the difference between the three, is this a usage difference between American English and British English? What is the difference?
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Based on what Wikipedia and LDOCE suggest, the difference isn't between the American and British English (apart from the little bit of background on ditch). It's just the subtle difference in the type of hole that's dug/built.
Ditch and trench are much closer in the meaning, while gutter is slightly different.
A street gutter is a depression running parallel to a road designed to collect rainwater flowing along the street and divert it into a storm drain. A gutter alleviates water buildup on a street, allowing pedestrians to pass without walking through puddles and reducing the risk of hydroplaning by road vehicles.
Needless to point out that the rain gutter is an open pipe at the edge of a roof of a building that's used to collect and carry away the rainwater.
A ditch is usually defined as a small to moderate depression created to channel water.
In Anglo-Saxon, the word dïc already existed and was pronounced "deek" in northern England and "deetch" in the south. The origins of the word lie in digging a trench and forming the upcast soil into a bank alongside it. This practice has meant that the name dïc was given to either the excavation or the bank, and evolved to both the words "dike"/"dyke" and "ditch". Thus Offa's Dyke is a combined structure and Car Dyke is a trench, though it once had raised banks as well. In the midlands and north of England, and in the United States, a dike is what a ditch is in the south, a property boundary marker or small drainage channel. Where it carries a stream, it may be called a running dike as in Rippingale Running Dike, which leads water from the catchwater drain, Car Dyke, to the South Forty Foot Drain in Lincolnshire (TF1427). The Weir Dike is a soak dike in Bourne North Fen, near Twenty and alongside the River Glen.
A ditch can be used for drainage, to drain water from low-lying areas, alongside roadways or fields, or to channel water from a more distant source for plant irrigation. A trench is a long narrow ditch. Ditches are commonly seen around farmland especially in areas that have required drainage, such as The Fens in eastern England and the Netherlands.
A trench is a type of excavation or depression in the ground that is generally deeper than it is wide (as opposed to a wider gully or ditch), and narrow compared to its length (as opposed to a simple hole).
So, as it's comprehensible, a ditch is usually deeper and wider than a gutter, whereas a trench would be usually used in civil or army terms, in addition to the geological meaning that it has.