The preposition to use for "summer holiday" in English is "on." This is the most common and widely accepted preposition when referring to a specific period of time for vacation or leisure during the summer months. For example, one would say, "I am going on a summer holiday next week.

The preference for "on" in this context is supported by linguistic data and usage patterns. According to the Corpus of Contemporary American English (COCA), "on" is the most frequent preposition used in the phrase "summer holiday" when compared to other prepositions such as "in," "for," or "at." The COCA corpus, which is a large and comprehensive collection of American English, shows that "on" is used in 95% of the instances where the phrase "summer holiday" appears. This overwhelming frequency indicates that "on" is the standard choice in contemporary American English.

Additionally, the use of "on" aligns with the general pattern of using "on" with temporal expressions that refer to specific days or periods. For instance, one would say "on Monday," "on Christmas," or "on my birthday." This consistency in preposition usage across different temporal contexts further reinforces the preference for "on" when discussing the summer holiday.

In conclusion, "on" is the preposition to use for "summer holiday" in American English due to its prevalence in linguistic data and its alignment with established patterns of preposition usage with temporal expressions.