When do we use "any" with countable nouns?

It depends which kind of any you are talking about.

The more common any is a negative polarity equivalent of some. Like some, it is used with a plural or an uncountable:

I have some books, some wine.

Do you have any books, any wine?

But we would not normally say

? I have some book.

or

? Do you have any book?

But there is a second use of any which is not a negative polarity item, and is used with countables, singular or plural; it means something like I don't care which.

You can choose any book / any books.

If you use it with an uncountable, it will be like using "a" with the uncountable, and designate a kind or sort:

You can choose any wine.

means "any kind of wine".

From § 2.54, CoGEL, "any", as a determiner, is used in the so-called non-assertive grammatical contexts. It can be found at § 5.14, Type (b) (ii) that it is the non-assertive central determiner of plural count nouns and noncount nouns (of a type called called quantifiers, § 5.23). Therefore, it is not considered acceptable to use it with singular count nouns. The meaning of any in this case is the counterpart of "some" (unstressed "some", which should be /səm/. In this context "any" is not stressed either.

Count

• They had some marbles in their pockets.
• Did they have any marbles in their pockets?

Noncount

• He drinks some soda with his meals.
• Does he drink any soda with his meals?
• Doesn't he drink any soda with his meals?

However, from § 6.61 we see that "any" occurs with a different meaning in assertive territory. This meaning is "it doesn't matter which/what/who". While the word is still a determiner, it is not now a quantifier and it is stressed.

Examples from CoGEL

• He will eat any kind of vegetable.
• Any dog might bite a child if teased.
• Any offer would be better than this.

This does not mean that it does not occur with this meaning in non-assertive territory.

• Did he tell you to read any book on the subject? No, not any, but only that one which has been recommended by the teacher. (user LPH's example)

This is the explanation for the occurrence of both plural and singular.