No I do not think there is a simplistic easy way to remember it is something that many natives acquire. You can use any with countable and uncountable nouns. These two connected links will give you plenty to study. 1: - https://dictionary.cambridge.org/dictionary/english/any 2: - https://dictionary.cambridge.org/grammar/british-grammar/any [ this page discusses some common errors] I recommended you study the common errors highlighted on each page. here is one page of those error warnings. Any: typical errors We don’t use ‘weak form’ any with singular countable nouns: They have a big lunch in school so they don’t need a hot meal in the evening. Not: … so they don’t need any hot meal in the evening. We use every not any with singular countable nouns when we mean ‘each individual member of a group’: The hotel is luxurious and it has every type of activity. Not: … it has any type of activity … See also: Every We don’t use ‘weak form’ any on its own in statements. We use no or we put a negative word, e.g. not, before it: There were no lifts so we had to climb five flights of stairs. There weren’t any lifts so we had to climb five flights of stairs. Not: There were any lifts … We use some not any before countable nouns in statements to refer to an indefinite quantity of something: I must buy some plants. They’re so nice. Not: I must buy any plants. You need to read the page by scrolling down to the bottom of the page on the second link, because Italki has removed the text formatting so you cannot see what has been crossed off with a strike through.

I think we just take it casual in oral practicing. According to Oxford Dic… When 'any' uses with uncountable or plural nouns in negative sentences and questions, it refers to an amount or a number of something, however large or small. 'Do you have any questions?'👉 stressing on the quantity. When it comes with singular countable nouns, it refers to one of a number of things or people no matter which one. 'Any color will do.' 👉which one doesn't matter

Hello, "Any" is used with countable plural nouns: Any + plural countable nouns Example: Did you see any people walk by? I can't find my friend. The exception is, of course, uncountable nouns: Any + uncountable "singular" noun Example: Do you have any water? Is there any sugar in this coffee? For the examples you have above, "question" and "problem" are both countable nouns (we can add -s to make them plural) so you must use any +____s: Do you have any questions? Are there any problems? There are words with "any" built in like: anyone, anything. These are obviously kept as is. Example: I don't know anyone who likes this book. I don't have anything to give him for Christmas.

From your examples, I'd guess the rule is: "Don't use singular".