Answer:
Some actors work as a waiter until their big break.
Step-by-step explanation:
The sentence that contains an inappropriate shift in person is the last one. Instead of a waiter, we should say waiters (Some actors work as waiters...). Actors is a plural noun and it should be followed by a plural noun. Another thing that confirms this is the use of the plural possessive pronoun their.
The rest of the sentences are correct. In the first sentence, there is no dilemma. In the second sentence, we have the phrase many people, which requires a plural verb and plural possessive pronoun. These conditions are fulfilled. The third sentence is a bit tricky because of the correlative conjunction neither... nor. The verb and possessive pronoun should be in the singular form in this case (as Dave is a singular noun), and they are, which makes this sentence correct.
This is why the fourth option is the correct answer.