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Skeptics requires questions to be about notable claims. The burden of proof is on the question to provide evidence that the claim is notable.

Must the question also provide evidence that there is a notable doubt about the claim?

This is particularly an issue for questions about cultural phenomena. What's unbelievable to someone from one cultural background might be so normal to someone from another culture that they never even stopped to think about it. Only in the last week I learned that droppings are not commonplace everywhere (not that I'm surprised, but the thought that they aren't simply never crossed my mind). Perhaps someone from a different culture may find this so strange that they are not sure if it's true.

For example:

What are the defining criteria here?

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    Don't understand the issue. The rule says "claims must be notable". And that is somewhat measurable: impact, reach, views, prominent person… But what do you mena with notable doubt? The asker has doubt, is a single person: must his/her doubt be notable? How should that be agreed upon? – LangLаngС Jul 29 '19 at 16:22
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    "Do American kids really sing to the flag?" If you mean the pledge, then no. There's plenty of songs about the flag and that reference the flag. Singing to the flag would be a first for me, an American. – fredsbend♦ Jul 31 '19 at 0:38
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    To be clear, American school children recite the pledge of allegiance. They do not sing it. I suppose it might sound a bit singsong, as it is recited quickly, by rote, every school day. But it's not actually sung. – Brythan Aug 2 '19 at 0:57
  • @Brythan I see, I thought they were signing (part of) the national anthem while the flag was being raised. My mistake. – gerrit Aug 13 '19 at 10:21
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I believe this is a pretty similar question to If a claim is commonly accepted, does questioning it require a notable counter-claim?

My proposed answer there I think applies here too - in summary:

TL;DR: We should only demand evidence of notability in one direction, and accept genuine disbelief of the claim by the OP as being sufficiently notable in the other.

Looking at your example questions:

  • Proof that the earth is round? was closed because the notability source was a humour site, rather than for being off-topic.

  • Is the Earth flat? was edited by two mods, and re-opened once by a mod (me), before being closed by 5 members of the community. Interesting that the mods think it is on-topic but the community doesn't. (I want to take a lesson from that, but I haven't yet.)

  • Is the sky blue? Under my proposal, the OP would be asked to why they are skeptical, to see if the question has been asked in good faith, and if so, should be on topic. [Aside: Using "The sky is blue" as an example of an obvious fact has always struck me as hilarious. It is often black, white, red, pink, yellow...

  • Do Dutch parents really drop their kids in the woods? Seems on-topic, but one where the simplest of Google searches would address it. [Aside: I would already be editing out the "really" before I had finished reading the rest of the question.

  • Do American kids really sing to the flag? It would be clearer if the question had a scope - ALL US kids? SOME US kids? In 2019?

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    I was very confused for a minute there. You put two links to "Proof that the earth is round?" but it seems like the second link should be Is the Earth flat. – JMac Jul 30 '19 at 13:42
  • @jmac: Corrected. Thanks. Too many open tabs, obviously! Sorry. – Oddthinking♦ Jul 30 '19 at 20:59
  • How you know there is genuine disbelief on the part of the questioner? One way of course is to assume that if the answer is widely known then they are not asking in good faith. This is a great approach, but doesn't it lead us to the same place? If there is no significant belief in the falsity of the question then we are allowed to close it. – DJClayworth Jul 31 '19 at 1:52
  • @dlclayworth: I have asked them in the comments. To date, this send to have got reasonable enough results (with answers in both directions). – Oddthinking♦ Jul 31 '19 at 4:18
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We have historically said that only the claim needs to be notable, and this has led to some questions that should have been closed, and would have been closed had they asked in the reverse sense. This is not good for the site.

The logic is simple. If "Is the earth flat?" is not a good question, then "Is the earth not flat?" is also not a good question for exactly the same reason.

Questions should only be on topic if there is notability on both sides - if a notable set of people believe both the truth and falsehood of the statement.

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    "The logic is simple." You'll have to spell it out then, because my understanding of logic is that not x = a is not the same as x = a. – fredsbend♦ Jul 31 '19 at 0:41
  • My point is that "Is x equal to a" is the same question as "Is x not equal to a" ,just with yes and no reversed. Whether yes and no are reversed makes no fundamental difference to the quality of the question. – DJClayworth Jul 31 '19 at 1:56
  • I agree that the two questions are equivalent, and have edited titles to reverse them to avoid clumsy negations. But having to prove that many people believe and disbelieve a claim, without putting the answer in the question, is pretty tough needle to thread. – Oddthinking♦ Jul 31 '19 at 4:21
  • OK, I see your point. We might have difficulty finding enough people aware of a claim to find a notable number both for and against.. But the claim Ms we want to target (I.e close) are the ones where there is a generally accepted answer (earth is round). If we could close those it would be progress. Maybe we need a little more nuance for questions where the claim is little known, but even if we were unable to close those closing the obvious ones would be progress. – DJClayworth Aug 1 '19 at 2:22

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