Whether “numbers of” or “number of” is correct will depend on the noun you’re counting (or not counting). Basically, if the word or context is plural, then you will use “numbers of” but if they’re singular, you’ll say “number of.” However, both imply a great many multitudes of an object.
What’s confusing about this is the –s and the fact that both phrases suggest plurality. Therefore, in some situations, either phrase can be interchangeable with one another.
You will use “numbers of” when referring to something in high quantities in the plural. This will accompany verb agreement where applicable. Consider the following examples:
There are huge numbers of people in the crowd.
When we arrived at the show, the numbers of people present were overwhelming.
Since “number of” suggests singular usage, you will apply it when there is only one noun in question. However, it will also suggest a kind of plurality. For instance:
I have told you a number of times that I don’t want to go camping.
No one can ever know the whole number of historical data in their lifetime.
Did you see the number of products they have at the fruit stand?
The number of crystalline salt grains is uncountable.
While you should use plural agreement when using “numbers of” or “number of,” they can interchange and still be grammatically succinct. However, for displaying a command of the language, it’s always good to ensure the noun and verb agree with the singular or plural in regards to either phrase.