Which of the following sentences is grammatically correct?

The sentence "Somebody left his or her jacket on the train" is grammatically correct because it appropriately matches the singular pronoun "somebody" with the singular pronouns "his or her." This maintains subject-pronoun agreement, which is essential in ensuring clarity and correctness in writing.

In this context, each pronoun corresponds specifically to the singular noun "somebody," emphasizing the individual nature of the subject. This construction allows for clarity in understanding that the jacket belongs to an unspecified person.

The other sentences either falter in pronoun agreement or use forms that are widely accepted in informal contexts but less so in formal writing. For instance, the second sentence incorrectly uses "several," which is plural, with "his or her," which is singular. The last two sentences use "everyone" which is singular. While "their" is commonly used in casual conversation as a singular pronoun to refer back to "everyone," it is not traditionally accepted in formal grammar.

Thus, the first sentence stands out as the only one that adheres to the rules of grammatical agreement, making it the correct choice.