Should nouns after "any" be singular or plural?

While I was answering a question on ELL, I got confused with a comment from a fellow user. After I indicated that:

"Do you have any idea to prove it?'

is grammatically correct, the OP commented that "ideas" should be used in place of "idea".

The comment befuddled me. I thought until now that "any" should be followed by singular noun and not plural. So I searched Google and saw it's a huge debate going on and there is no end to it. I got no perfect and trustworthy source. So I thought to post the question here.

So what is the right rule? Should singular or plural be used after "any"?

Answer

The correct use changes depending on the sentence:

Do you have any idea how to do this?
Do you have any idea what to do?
Do you have any ideas for me?
Do you have any ideas for how to do this?

It seems that, if the singular or plural noun (idea) is directly connected by a subordinating conjuction (how / what / where / which / that), you use the singular; if it's modified by a prepositional phrase (for...) or by nothing at all, you use the plural.

Without using all that grammar jargon, I'd say use the singular if it's followed by how, what, that, who, whom, whose, which, etc, and use the plural otherwise.