I was just wondering what is more commonly used in Japan, in terms of formality? I feel like I remembered reading somewhere that Genki: 3rd (or 2nd?) Edition made note that じゃないです is more commonly used colloquially than じゃありません (ie. talking with a server at a restaurant “formal” vs. business “formal”), is this true? Or are both equally as acceptable?
I’m not looking to make any business relationships lol so I just want to make sure I’m approaching a situation the right way.
In terms of everyday polite spoken Japanese, I would say じゃないですか is the most popular. However in terms of what I actually heard the most, that would be じゃない・じゃん . It actually took a while to understand what じゃん meant but once it clicked that it means that same thing as the other two if not a little more playful.
Some examples
Is it ok?/It is ok
いいんじゃないですか ・いいのです
いいんじゃない？・いいよ
いいじゃん？・いいじゃん！
I think what @dylenmg is asking is that they’re both polite forms, so which is acceptable?
I’d say both are “normal” polite level (not business polite). じゃありません sounds just a touch more polite.
Polite
車です
車じゃないです
車じゃありません
Casual
車だ
車じゃない
Notice that casual still uses じゃない so adding です at the end is a sorta “quick fix” way to make it polite – you can suddenly at the end change from casual to polite. But when you say じゃありません you’re definitely heading into the ending with the intent to be polite. 
You know how Japanese people tend to soften everything and rarely express direct negation. They try to be polite and use words that carry a softer tone, most of the time.
じゃないです is technically wrong, as the polite version of ない is ありません. However, there are two reasons why it is used much more than ありません in daily life.
1- It’s faster and sounds less formal (strangers aren’t demon kings, you can speak relatively casually)
2- It’s softer. じゃないです consists of a casual negation, with です added to make it more polite.
じゃありません , despite sounding more formal, is very direct and carries a strong feeling of negation. Pay attention when using it, as it could sound a bit direct and possibly out of place if not used in the right situation.
If you’re speaking to someone of higher status, you can always use ではないです to sound more polite, but softer than ありません. (じゃ is a contraction of では, in case you didn’t know)
I shared this on WK a few months ago in a similar topic, the studies show more depth of tendencies depending on context/situation, writing vs speech, declarative vs interrogative and even tendency by age showing in evolution in patterns that have changed through the years.
This is brilliant, great work on your thesis. It’s very thorough And well thought out!
It’s not my thesis, just a reference I found 
I can’t speak to any theory behind which to use when, but in personal experience, 「じゃないです」 is overwhelmingly more common. Every time I say 「じゃありません」 to one of my coworkers (who’s my superior in both age and position) she lovingly replies with 「おばあちゃんだ」。The only time I can comfortably crack out 「じゃありません」 is around the school principal and people older than 60.
In terms of business I think it’s an interesting line to walk. So much of the language is built around balancing speaking politely with speaking approachably.
じゃないです is far more commonly used and is still considered polite. じゃありません is more formal than polite and much more direct. Also, never use じゃん with someone unless you are very close with them, incredibly drunk with them, or they’re your subordinates. 
じゃありません is mostly used by people who learn Japanese from Minna no Nihongo /s
buuuurn