His cloak was wet due to the heavy rainfall
His cloak was wet due to the heavy rains
His cloak was wet due the heavy rain.
Which out of the three sound more appropriate? Is the use of the article 'the' required?
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The article "the" is not required in this context. You could just as easily have "due to heavy _" in all three sentences.
The third sentence is wrong, because you have missed the word "to": Due to the heavy rain.
As for which sounds more appropriate, it doesn't matter. In this context, "rainfall", "rains", and "rain" are interchangeable. In general, the word "rains" refers to multiple showers, while "rain" and "rainfall" can be one or more showers.
I agree with @AricFowler the definite article "the" is optional here.
All the three words rainfall/rains/rain are actually synonyms. Notice that in British English the rains refers to the season when there are heavy rainfalls.
As per definition, rain is water that falls from the sky. Thus, if you see water falling from the sky, you can refer to it as rain.
Rainfall is used when we want to communicate how much rain falls in a certain area over a specific period/day/time or in a more general way, the average amount of rain that falls over a specific period to communicate overall rain status.