Context → patterns of usage section for 人気: https://www.wanikani.com/vocabulary/人気
何で「人気メニュー 」ですか。
As an な-adjective, should it be: 「人気なメニュー」？
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It’s also a の adjective, which is how it’s used in this example, and の can be omitted. So, I’d say, both are possible…
人気メニュー is not a menu that is popular. It’s a list of popular dishes.
Like a サイドメニュー (side menu) is a list of side dishes, and a breakfast menu is a list of breakfast dishes. “Breakfast” is used as a modifier here but it is not an adjective. 人気 is being used in the same way.
I’m not a native speaker, but to me 人気・な・メニュー sounds like “a menu that is popular.”
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Thanks to both you and @trunklayer for your insights here. They do break out 人気のメニュー as a separate example and translate it slightly differently (popular menu items) vs (popular menu) for 気メニュー. It feels to me that maybe the small distinctions here are sometimes more subtle than can be translated via English and also related to the context of usage.
A final grammar question: it seems that の in applying adjectives is often omitted in casual speech, is that also the case for な in な-adjectives?
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Unlike the の particle between a noun and the word it describes, I’ve never seen the な omitted between the な-adjective and the word it describes, but that doesn’t mean much since I still have a lot to learn 
I’ll second this - both not being used to seeing な omitted and not feeling quite experienced enough to rule it out.
Looking at the “Patterns of use” examples here on WaniKani, I’d argue that 人気のメニュー and 人気のひみつ aren’t really examples of 人気 as a の-adjective, but as a noun; it’s a “popularity menu” and “popularity secret”.
That being said, there is an argument to be made the term “の-adjective” is a bit deceptive to begin with, even if it can be useful; they’re essentially more like genitive noun constructions, but used in a descriptive sense where they are commonly translated as adjectives in English.
人気のスポット could be translated as “popularity spots” (or “spots pertaining to popularity”), but that doesn’t sound good in English, so we call them “popular spots”, and thereby think of 人気 as an adjective.
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I have something to ask/share on this topic. Sometime ago, while studying here, I noticed that adjectives marked as both な and の have examples only for の. I asked my teacher to comment, and she told me that an adjective cannot be both な and の at the same time. Here, I see that 人気 can act as both. So, I’m curious if she gave me a rule that has some exceptions or maybe I just misunderstood her. I’ll bring up this topic with her next class.
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