Final answer:
Photographs from a state fair in 1924 would be the best primary source to consult to learn about mustache styles from the 1920s as they provide visual documentation of the era's popular styles.
Step-by-step explanation:
To learn more about mustache styles that were popular during the 1920s, one should consult Photographs of people at a state fair in 1924. These photographs are likely to capture the general populace at a public event, showcasing a variety of mustache styles prevalent at the time.
Since they provide visual evidence of how people actually styled their facial hair, they serve as a more direct and reliable source than a population census, which would not include specific details about facial hair, or a magazine article about a contemporary mustache tournament.
Personal letters from Charlie Chaplin could offer some insights into the thoughts and trends of the era, particularly if they discuss personal grooming or public attitudes towards mustaches, but they would be limited to his perspective and might not provide a broad understanding of popular styles. Thus, for a general overview of the mustache styles of the 1920s, photographs from that era are the most informative primary source.