Although I know that the following is correct:

"There are three dogs"

I also see people writing/saying:

"There is three dogs"

Is it also correct?
No it is not correct, unless "Three Dogs" is the name of something.
"Oh, there is Three Dogs!" In this case Three Dogs would be a person or animal's name.
Students: We have free audio pronunciation exercises.
Using "is" when "are" is the correct word is a very common error these days.

I hear it constantly from news readers on tv and radio, people who should know better.

So how will kids or students of English ever learn it correctly?

They do not say "there is" when it should be "there are",
but they do say "there's" when it should be "there're".

e.g., There's three dogs in the garden. (is wrong!!)

Why do they do this? Simple laziness.
It is easier to say "there's" than to say "there're".

"There's" does not sound as obviously bad as "there is" so they get away with it,
but it really is just as bad as saying the words separately.

I notice it every single time, and it drives me about half mad. Emotion: surprise)
Yes, it is sloppy English. I often hear this type of mistake on both CNN and the BBC!!!
Hey Mike!

Yes. I hear it everyday, everywhere!
And yes, even on the BBC World Service radio shows!

Another pet peeve is the word "asked".

I think that "asked" is the most mispronounced word in the USA.

People almost always say assed, eliding the k,
or aksed, switching the s and the k.
Again, out of laziness.

Will we all be using digispeak one day? Emotion: surprise)
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To sum up, you were right to trust what you knew to be correct. 'There is' is for singular use, 'there are' is for plural use.
Bad English is common.
Our mission, should we decide to accept it, is to help preserve this wonderful international language from linguistic vandalism - lest confusion reign.