I think that I’ve noticed that polite negative verbs can be expressed in two ways: in the negative masu form and the negative casual form + desu.
Examples:
食べていません vs 食べていないです “to not be eating”
じゃありませんでした vs じゃなかったです “was not”
Is this correct? If so, are there any differences in meaning or usage between these two forms?
To my understanding ないです and ありません both mean the same thing and are both polite. I found a Japanese person saying that ないです is not technically grammatically correct (debatable) but people (especially young people) use it all the time so it’s one of those “language changes with time” kind of things. It is however considered a biiiit less polite than ありません by some.
There are probably rules like “don’t use ないです when writing academic text” or whatever but I can’t say for sure.
Looks like someone wrote a whole dissertation on this topic but I just grazed the abstract. According to this, there seems to be tendencies based on age group and certain particle endings. Speaking wise, I think it’s just easy to fall into plain patterns and you forgot the polite circumstance…so you land it with a desu quicky to make up for it.
Here‘s another (much shorter paper in Japanese) with a summary/partial translation on stackexchange:
indicating that 〜ないです is a softer negation with more focus on the politeness while 〜ません feels more assertive (in spoken Japanese)
This is literally exactly what I do. I’m so used to speaking casually that when cashiers ask me things like if I have a point card what naturally comes out is ない and then I just add です to it. It’s also what I hear most other customers around me use when asked though.
This is the first thing I thought of when I read the question; how I always respond with ないです when asked for a point card. 
Hello, essentially です is just an ornament you can drop at the end of a logical sentence to make it formal, or “elevate” it. Verbs have ます plopped at the end of themselves once they’re transformed into their i-stem instead.
I am aware that it is rather specific to use short form with the combined use of です. You are purposefully being informal towards someone but yet recognizing they are above you. So, you choose to still keep a degree of formality. Something very common is ないです。 Natives argue about things like this all the time, claiming that they’re ungrammatical or “no one really says it”, but it takes just a moment to run into someone who does. Remember that a native might not be able to give you a proper answer.
ないです sounds like if somebody got nervous because they mistakenly talked in a casual way to somebody older and try to correct themselves.
Like: いやいや、あのイカを食べなかった…ｄｄです（汗）
However, adding a ん in between makes it correct, and counts as polite speech.
あっ、すみません…英語わからないんですが…
Some people here said that everyone says that ないです is said to be incorrect, but yet Japanese people use it.
That’s probably either verb + ないんです or 無いです (as a sentence on its own) that is being used.
A：ゴミ箱ありますか？
B：無いです。
At least, that’s my observation around here.
I think they are actually saying ないです。without the ん in there. You can even here Teppei from Nihongo con Teppei say it. Also, it often doesn’t serve the same grammatical purpose as ないんです does, ie explaining, observing.
In certain contexts it is probably better to stick to ありません/ません, like when making a speech or something, but in everyday situations you can probably get away with ないです a lot of the time, like the Japanese do, too.
I don’t see why anyone would argue that verb-ないです is “incorrect” unless they also think い adjective + です is “incorrect.” There may have been a time when it was, but we are well past that.
Verb-ないです is normal, you hear it all the time, and quite useful.
What’s funny is that native linguists don’t seem to agree on this topic.
The source I posted mentions this contradiction from authors mentioned in SE (Tanaka, Noda) and then compared another linguist (Kawaguchi) and stated:
Kawaguchi showed that the shift from masen to nai desu forms occurs more in declarative sentences than in interrogative sentences by using natural speech material. In the case of declarative sentences, the shift to nai desu forms is more likely if the emphasis is on negation. This also seems to be true in interrogative sentences, especially for ‘negative proposition’ type sentences where negation takes emphasis. (p 40)
There were a couple tables that caught my attention that were interesting.
Here is a variety of nai des/masen ratios across several studies:
Here is another on particle ending variations with parts of speech
I didn’t read into all the hypothesis’ details (kinda just skipped around) but several revolved around Kawaguchi’s claim that nai desu is when denoting negation (results agreed), nai desu preferred for yes/no questions or repeated (results agreed with some exception), nai desu preferred for interrogative sentences (agreed, but not significant). There was another one about nai desu being used more in declarative versus interrogative but sort of a mixed result.
Of course not saying one is better that the other (have no idea), but there are clearly differences of opinion. With that, not sure what to take a away as a language learner other than it’s nice to have some forgiveness in grammar choices or least some personal choice in expression.
Yes… You’ve got yourself good answers now. Someone else pointed out something that I forgot to do in my comment, but nai is an adjective and desu allows it so that it takes adjectives too. So it is perfectly fine
One thing you have to keep in mind is that linguistics is descriptive rather than prescriptive. The fact that it’s in dispute means that enough people use it differently that:
Exactly. 
Then again, if you’re writing an academic text, you’re probably not using です・ます in the first place!
Yeah you’re right. Maybe I should’ve said “written language” instead of “academic text”. But then again, I really don’t have enough knowledge when it comes to written language and it’s different rules.
This page about halfway down mentions that い-adjective + です is incorrect in the consciousnesses of some older speakers. Probably for the same reason today nearly everyone will say い-adjective + だ is incorrect: い-adjectives don’t pair a copula within the logic of the Japanese language. When です is added, it is not to function as copula, since い-adjectives already have a copular function. です is not a particularly old word in Japanese, but from its inception it was a form of the copula. Whereas, adding です to an い-adjective is purely for politeness. But, this means that です, which was derived as a copula and nothing more, is being given a new function totally unrelated to coupling, and one that is not included in the etymology of the word.
In short, there are two different ですs. One is a copula, and the other is a like a polite sentence ending particle. If I understand correctly, final particles in Japanese at one time (perhaps very far back) carried a more concrete grammatical character. Japanese has made room for these sort of “garbage” syllables that over time flock toward the end of a sentence (or even clause) in order to convey subtle information in a semi- or non-grammatical way (that is, instead of spelling out the subtle implication using verb, noun, adjective, or adverb grammar). For this reason I think you can ignore concerns about the illogic of -いです . Just recognize that, grammatically, it’s two different uses for です. Like how the particle わ is apparently derived from the topic particle は, but no one would claim they represent the same word.
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