Teo wrote:

1. Do you have a child? 2. Do you have any child? 3. Do you have any children? 4. There isn't a hospital in our town. 5. There isn't any hospital in our town. 6. There aren't any hospitals in our town. Sentences #2 and #3 are incorrect. Do you agree?

I'm no expert but here's my take on this:

1, 3, 4, 5 and 6 are correct.  2 is incorrect.

"Any" and "a" can mean the same when "any" implies "one and only one."  E.g. "Pick a card."  or "Pick any card."  Both mean the same thing.  "Pick one card."  "Here is a list of hospitals for your insurance plan.  Pick any hospital."  It implies that you are to choose one and not one or more.

When "any" does not imply "one and only one," "a" and "any" are not interchangeable.

"a" goes with a singular countable.  "Any" goes with one or more countables (plural form) or when talking about the existence of an uncountable.

"A cow (singular countable)."  "Any cows (plural countables)."  "Any water (uncountable - does it exist?)."

"There are" is used for plurals.  "There is" is used for singulars and uncountables.  Countables are preceeded by "a" if we are only talking about one.

e.g. "There is a cow." 

Uncountables and plurals need nothing.  "There is water."  "There are cows."

"Any" means "one or more" when talking about countables so you can substitute that phrase for "any" and see if it still makes sense. 

#3 - "Do you have any children?"  becomes "Do you have one or more children?"  It still makes sense.

"Child" is countable (specifically one) and it doesn't make sense to say "Do you have one or more child?" because "more" can mean two whereas "child" only means one.

"Any" means "existence or presence" when talking about uncountables.   "Is there any water?"

"not any" means "zero" or "no" so it depends on what "any" is describing that determines whether you use "There aren't" or "There isn't" for your sentence.

You can use "There isn't any" when describing something uncountable like water or air. "There isn't any water in the pool." or "There isn't any air in the room."  If it is countable and you're talking about one, you use "There isn't a"  or "There isn't any" because "not a" and "not any" both mean "zero" or "no."  "There isn't a cow in the field."  "There isn't any cow in the field."  Use "There aren't any" when talking about countable plurals.  "There aren't any cows in the field."

#2 is wrong if you mean to imply "one or more." You don't usually use "any" with child because although a child goes with "one" it doesn't go with "more than one."    You could use any with child if you were implying "only one" like my card example.  E.g. "As a Big Brother, you can pick any child from the program to mentor."  The word "child" specifically means one kid so if the question is meant to imply "one or more", use "any" with "children" instead.  "Do you have a child?" or "Do you have any children?" Children is plural so you never use "a."

"There aren't any hospitals in our town."  (plural countable)

"There isn't a hospital in our town."(singular countable)

"There isn't any hospital in our town."  (singular countable, "not any" means no).

"There aren't a hospital in our town."  (wrong because hospital is singular/countable, use isn't not aren't).

Teo wrote:

Grammatically conceivable, but nevertheless still unacceptable on grounds of extreme unnaturalness. We do not normally use 'any' with an unmodified countable noun in a simple interrogative of this type (#5, you will observe, is a negative, not interrogative sentence, to which somewhat different restrictions apply). It is an instance of the kind of idiomatic restriction which only an extensive experience of English can teach you. My advice to you,in case of doubt, is not to attempt to use 'any' is such contexts. A simple indefinite article will rarely, if ever, be incorrect(Do you have a child? There isn't a hospital here., etc.) <removed>

Paco2004 wrote:

Once I was blamed here that I used Google too much, but I still believe a google search will give us a good clue about what form of sentence a majority of English speakers actually use for such a questioning. Do you have children? 2,040,000 Do you have any children? 185,000 Do you have a child? 86,300 Do you have any child? 295 Do you have pets? 159,000 Do you have any pets? 447,000 Do you have a pet? 125,300 Do you have any pet? 149,000 The results agree with my understanding that "any X" has a qualitative sense (= "any kind of X") when X is a countable noun, whereas "any Xs" can imply a quantitative sense (i.e., "any number of Xs") in addition to the qualitative sense. paco

You can't just do a search in Google and accept the count as the true result.  For example if I search on the following: "Do you have any pet?", I get a results that include "Do you have any pet frogs?" or  "Do you have any pet peeves?"   Those results are correct english but have nothing to do with the search result you are interested in. 

Google doesn't search on punctuation so it doesn't know you only want that phrase without any additional words tacked  onto the front or back.  As a consequence, those are included in your search results.  So your results are going to be greatly skewed upward for that particular search.  There may be only one or two results that are used in the manner you think you are searching on.  The rest have additional words which don't relate to your search..

By the way, when searching for a phrase, surround your search phrase with quotation marks so you'll only get relevant hits.  When I search for the following:

"Do you have any pet?"  I only get 13,900 hits.

Paco2004 wrote:

Hi RRJ Thanks for the advice. I have to admit the result of my previous search was a bit skewed and misleading. I did a new google to check it. "Do you have pets? I" 1130 "Do you have any pets? I" 1570 "Do you have a pet? I" 1130 "Do you have any pet? I" 35 So the new search suggests "Do you have any pet?" is of rare use. I believe google search is a useful tool for us ESL students to know current usages of English, if we use it properly. We can learn something about English through talking with native speakers in places online like here, but I have a feeling that 'an English' that a native speaker or any person who believes he/she is a good English speaker speaks or hears personally is not always the same as the English spoken by the majority of English speakers. Google search is a good tool to check whether an expression such an individual person asserts to be colloquial is really spoken by the majority. paco

That's a clever search and just for clarity's sake, searching on, 

"Do you have any pet? I" and "Do you have any pet I" 

with or without the question mark is the same to Google

Google is probably a good way to see how most people write English as opposed to how they're supposed to write English.  Just know that there is plenty of bad grammar out there with plenty of improper word use (e.g. They're, there, their or it's, its).  Another classic example are slang words like "ain't."  Blogs would probably give you a good idea of English word use too.