All of the sentences contain both a noun phrase and a verb phrase. However, if you need to identify one that clearly demonstrates both:
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In "He sat on my grandmother’s favorite armchair," the noun phrase is "my grandmother’s favorite armchair," and the verb phrase is "sat on."
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In "The pigeons have not landed on my neighbor’s balcony in a long time," the noun phrase is "The pigeons," and the verb phrase is "have not landed on my neighbor’s balcony."
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In "Some dog breeds have docked tails," the noun phrase is "Some dog breeds," and the verb phrase is "have docked tails."
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In "You can find frogs and insects by the pond if you look closely enough," the noun phrases include "frogs and insects" and "the pond," while the verb phrases are "can find" and "look closely enough."
So all the sentences meet the criteria. However, if you are looking for just one example, any of these sentences would suffice.