Is this correct?

Not really, but it’s a mistake you’ll see made sometimes. Because ‘results’ is plural, it should be ‘there are not many results’. Even better is ‘there aren’t many results’

"There aren't very many results".

It depends, I’m from Ireland and live in the UK and this type of phrasing sounds completely normal. For the US/Canada I don’t think it would work.

I’m from the UK and this is exactly how I would say it

No, but people talk like this a lot.

Not incorrect. However unnecessary. You're adding words that you don't "need" to say to convey a message. You can say for example, not many results found. or few results found, to get the message across. However, if you want to give emphasis to your sentence using those words in a sentence is considered "acceptable" because you conveyed the message correctly. Above all else that matters more than you accidentally changing the topic by saying something that has nothing to do with the topic of discussion.

The writer gives the game away with the second sentence . "Can you show *them* to me please?" Plural.

It’s dialogue, someone is speaking it. People don’t speak in complete and udder accordance with grammar conventions and that is why we see evolution in language. What matters here is that you understood what was being said, if you couldn’t then there was a failing on either the writers to communicate, a failing on your part to interpret, or a mix of failing on both sides of the equation as is most typical.

Technically, you're supposed to say "There are" or "There aren't" if you're talking about multiple things, but it's *extremely* common to use "There is" regardless of if the thing you're talking about is singular or plural

In formal and academic writing or scripted formal/academic speeches, no. Otherwise, yes. This is an extremely common way to speak, although some dialects may retain a more rigid distinction between “there’s” and “there are”. Native speakers do this regularly, and it is not wrong in normal conversation and writing, or even in formal unscripted conversation. Just don’t use it in school or in formal writing and you’re good. Learners get formal grammar rules hammered into their heads by teachers who don’t really want to acknowledge that most English is not in perfect prescriptivist grammar. Languages change and evolve and spoken English is a lot less rigid than written formal English or academia. One thing that’s important to understand is that native speakers aren’t speaking based off of memorized grammar rules; they’re speaking based off of what they hear and have been exposed to, allowing them to intuitively know what does and doesn’t sound natural, regardless of what the academic/formal written rules say. That’s not based on natural speech, but rather a prescriptivist idea of what the language *should* be. That’s fine for formal and academic writing (because it provided a standard that allows for wider understanding since dialects can vary drastically), but you really can’t expect natural speech, even in formal settings, to align with it perfectly because that’s not how language works.

"there are not many results"

I notice this all the time -- people use "there's" with plurals pretty often, but to me it doesn't sound quite right. In formal text I would definitely correct it to "there are"

It's a normal way people talk. I actually didn't notice there was anything wrong until I read these comments. I would say a lot of times, we reanalyze something as being a singular concept to justify "there's". Especially when there's a lot of words ahead of the noun itself like there is here. "very many results" becomes like, a cohesive idea that is one singular thing

So " very many " Is a acceptable use of words are there any more examples?

It is *not* grammatically correct. But it *is* an accurate depiction of how such a statement would be made in spoken English - "colloquial". So, it's kind of tricky. The grammar is bad, but as this is written dialogue, it can be considered "correct" if it is in character for the speaker to use colloquial speech.

it's quoted so it's dialogue. It's technically wrong but accepted under descriptivist grammar. "There's" tends be the default for a lot of people when they're speaking.

Technically, there are no grammar rules for speech. Grammar rules can only to written text. You can say anything, no matter how grammatically incorrect, and you haven't broken any grammar rules. The speaker is grammatically incorrect in the usage of "is not" to be correct they should use "are not/aren't" as the word "results" is a countable plural noun.