Learn Spanish (Latin America): Nadie sabe el secreto.

You only need a double negative if the negative word comes after the verb. Because nadie (nobody) is placed before the verb sabe, you do not need to add a no. If you wanted to put nadie at the end of the sentence, it would become No sabe el secreto nadie.

Since nadie means zero people, why is the verb sabe in the singular form?

In Spanish, indefinite pronouns for people like nadie (nobody) and alguien (somebody) always take the third-person singular form of the verb, just like él (he) or ella (she). Therefore, we use sabe instead of the plural saben.

Why do we use sabe instead of conoce for "knows" in this sentence?

You use the verb saber (conjugated here as sabe) when talking about knowing facts, information, or how to do something. A secret is a piece of information. You would use conocer if you were talking about being familiar with a person or a place.

I remember that nadie sometimes needs an a before it. Why isn't it A nadie sabe here?