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.
When you use the word zero as a number, the word it quantifies should always, I repeat, always be plural!
Example #1:
Ice melts at zero degrees Celsius.
Example #2:
— How many friends do you have in this town?
— After that story went public, I have literally zero friends!
You may ask why is that true? Well, consider this. You can have three cars, you can have two cars and you can have just one car. But how many cars do you have when you don't have any? Notice, I said how many cars, not how many car. So, naturally, your response should be I have zero cars.
The determiner many always implies plurality unless you have only one of something. If it was indeed possible to say zero car, then your response would have been either I have a zero car or I have zero car. Well, the first one sounds more like you're talking abut a type of car, not how many cars you've got. And the second one sounds like a name or title (the name of a game or a movie title, perhaps). Do you see the confusion?
The same holds true when talking about temperatures. The word degree should always be plural unless it's exactly 1 degree Fahrenheit or Celsius. Even when you're dealing with a fraction of a degree, it's still plural.
Example:
The outdoor temperature is 0.2 degrees Fahrenheit.
PS: The question how many cars do you have? would be more likely answered like this:
— How many cars do you have?
— I don't have a car. / I've got none.
The only time you ever say I've got zero cars is when you want to emphasize the fact that you don't have any cars at all. So, it's just a more emphatic way to stress that.
Here's some more information on the subject you might consider reading: https://english.stackexchange.com/questions/38293/why-is-zero-plural
In summary, 1 or -1 degree Celsius should grammatically always be singular, but in everyday speech a lot of people might say 1 or -1 degrees. Here's the rule: if exactly one of something—singular (regardless of whether the quantity is negative or positive). Everything else—plural (regardless of whether the quantity is negative or positive).
Best Answer
There is no general rule, if singular or plural should be used after no/any. It depends on the situation and context.
For your first example the correct sentence would be:
Edit: Because any is used in front of nouns to refer to indefinite or unknown quantities or an unlimited entity, all countable nouns (e.g. apples, chairs, etc.) are in plural and all uncountable nouns are in the "singular" (e.g. money, bread, etc.).
For your second example, either sentence can be correct, depending on the context.