This is a very soft question, that is inspired by an edit to one of my questions, which involved a change of the capitalization in the title; the details are not important and I don't mind such kind of editing at all (no whining intended).
I know that there are different accepted possibilities among "grammar lawyers" which one could check here, but I would like to hear about other people's preferred capitalization for titles of MO questions.
Both systems of capitalization are OK for use here:
How to Answer Questions on MO.
How to answer questions on MO.
Conclusions:
Which system should you use? The one most comfortable to you.
Should you change someone else's post from one system to the other? No.
Should you complain if someone else changes your post from one system to the other? No.
A title of an MO question should provide some information on the content of the body of the question post. At a maximal length of 150 characters they can be relatively long and it is encouraged to write informative titles.
A way to achieve this can be to reuse the main question of the post, or a modification thereof, as the title. Indeed, there is a recommendation to make your question your title in the general guidelines on asking. (Tangentially, this does not mean this question can be omitted from the body of the post; the body of the post should not depend on the title.)
However, this is just a suggestion and any title that is informative should be fine.
I am not certain what this implies on capitalization. But I think that at least if the title is long to use standard capitalization is more common and likely preferable to capitalizing each (relevant) word. I feel standard-capitalization is what is predominant in the titles of scientific papers, too.
But I do not think it is much of an issue either way. Only, I feel as a rule titles should not start with a lower-case word, as often this looks a bit sloppy in my opinion.
I suppose that all of us who don't have English as mother tongue would rather not have the idea of capitalizing nouns or other "key words" in titles. At least to me that idea has never occurred for a question in MO (even though it is the general rule for nouns in German, my mother tongue! - which is in my opinion somewhat obsolete).
It is about questions, not about article titles. And as I skim through the MO questions, only a minority seems to capitalize the titles anyway.
This being said, proper names should of course be capitalized systematically, also in the text. It is "Galois fields", not "galois fields". OTOH, for adjectives like "noetherian" or "gaussian", I see that many people don't capitalize. I personally would prefer to do it.
I think there are two different ways of writing a title for any given MO post: Question & Non-Question style.
Any title in one of these two styles should be chosen in such a way to be both informative and attractive for a typical reader in the first view. These two parameters (i.e attraction and information) are what I personally pay great attention to while choosing a title for my posts not really the capitalization of the words.
If you know how to use the keywords in your title, you will have no problem for coding enough information in short attractive phrases in both question and non-question styles. Correct capitalization can help in this direction. However it is a bit marginal with respect to choosing the right words for the title.
The main point is that your title is a kind of abstract for your question. While too short and generic titles are not that informative, too long ones are also not attractive and a bit hard to read and grasp in the first glimpse.
Anyway you should think and find the appropriate style and the optimum length for any of your questions to maximize the information and attraction in the title no matter whether it is written in a question or non-question style.
For example there could be two alternative ways for titling this question.
1- Examples of Common False Beliefs in Mathematics
2- What are examples of common false beliefs in mathematics?
Both of them seem attractive and informative enough to me to make me curious to open the post and look inside.
Moreover let me add the point that apparently there is no unique way of capitalizing the titles in English grammar. See Rule 16.a and Rule 16.b in this source. You can follow more precise descriptions in standard style books, if you want a pedantically comprehensive answer.
Though, I personally very much doubt that such level of attention to the pedantic points in English writing should be a typical MO user's priority when adding posts here, particularly when most of them (like me) are non-native English speakers who need to focus on improving their more essential linguistic abilities like expanding the domain of their vocabulary or mastering in daily used grammar to soften their fairly mechanical English.
