"There is" or "There are"
Which is correct to say - There is plenty of books or There are plenty of books.
And another examples - There is a lot of books. or There are a lot of books.
Which word is the main? If "Plenty" , "A lot" are the main words then it is correct to use singular that means "There is"?

Question

"There is" or "There are"
Which is correct to say - There is plenty of books          or        There are plenty of books.  
And another examples - There is a lot of books.       or         There are a lot of books.
Which word is the main? If  "Plenty" ,   "A lot"  are the main words then it is correct to use singular that means "There is"?

Answer

The verb has to agree with the noun. 'Books' is a plural noun, so you should use a plural verb:

How many books are there?

There are a lot of books.
There are plenty of books.
There are many books.

You use a singular verb with uncountable nouns:

How much information is there?

There is a lot of information.
There is lots of information.
There is plenty of information.

Answer

the proper use of there is and there are has to do with whether or not the subject is singular or plural.
There are is used for plural and there is used for singular. 
ex. There is a book - singular/ ex. There are some books - plural
Using "plenty" is mostly the same rule.