When do you use the following phrases?

a large number

large numbers

Are they different in usage?
You need to give us more context. I suspect that in some sentences either one could be used; in others there would be a difference.

Can you provide complete sentences with these expressions so that we can have a better idea of how to help?

I would use are in both sentences; otherwise, they are both fine. A large number of / large numbers of are quantifiers in my opinion.
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The simple contextual sentences are like this:

There is a large number of Chinese.

There are large numbers of Chinese.
 Mister Micawber's reply was promoted to an answer.
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Thank you, MM, but I was looking at another post, and one of the people replied said that "number" is a singular collective noun and it deserves to be follow up with "is" and not "are." But as it seems that the other case did not have "large" in front of number though, so the fact there is an adjective "large" makes the sentence to have a plural verb, "are." Right?

Some examples to consider:

There is a number of items to be scrutinized. (?)

There are a large number of items to be scrutinized. (?)

It is not a singular collective noun (team, family, staff, etc); it is a quantifier.

'A large number of is used before plurals, and a following verb is plural.'-- Swan, Practical English Usage.