The answer is (is), because it requires a singular verb.
Identify the two nouns that make up the compound subject in the following sentence.
"Neither the plates nor the serving bowl goes on that shelf."
The two nouns that make up the subject are "the plates" and "the serving bowl."
Neither the plates nor the serving bowl (go, goes) on that shelf.
goes, because the "serving bowl" is the part of the subject nearer the verb and it is singular, so the verb must be singular. They are also connected by "nor."
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