Looking back, that question was almost whimsical. It is a window into a different time in my life, when I haven't decided what to do for my research career (in fact, I had only recently decided to do a PhD in number theory at the time, and started working on a masters thesis of my own choosing. The masters thesis ended up being almost completely disjoint from my subsequent doctoral work). 

Since taking that first step, I have asked over 500 questions. Some of them have been very well-received, including one that earned me the Famous Question badge, awarded only 1376 times at the time of writing. Others have been quite poorly received, with negative scores. 

You will note that the two examples I gave explicitly so far have little to nothing to do with my research, as far as my publication list would indicate. This is a point I wish to emphasize: the main ingredient of a good MO question is not "depth" or relevance to your own research (which, to be honest, is probably quite niche): it is simply curiosity and a desire to engage with the unknown. 

However, we are in the (un)fortunate situation that we belong to a rich subject with a lot of history and existing literature. Having some engagement with the literature and understanding backgrounds is a necessary step before being able to fruitfully engage with the mathematical community. As such, even simple, innocent questions which are driven by curiosity can be deemed unacceptable at MO because it is believed that a reasonably well-trained mathematician, even in their beginning stages, ought to already know the content. 

Nevertheless, I would say that on average (in fact, an overwhelmingly positive proportion of the time), a question motivated by genuine curiosity and a desire to learn will be well received at MathOverflow, and the mathematical community at large. After all, most mathematicians are driven by the same basic instincts. 

Engaging with MathOverflow as part of your research program

The examples I chose above are of course not typical (at least not for me). The vast majority of questions I have asked on MathOverflow are, in fact, related to my research, and many of them intimately so. There is this question that I asked on 06 Mar 2012:

Some restricted weighted sums

and answered, to my surprise, by the illustrious Richard Stanley whose books I had studied vigorously for some time. Stanley's answer solved a technical conundrum in my attempt to generalize the global determinant method to the setting of weighted projective spaces, which ultimately led to my first paper: Power-Free Values of Binary Forms and the Global Determinant Method. 

If there is ever a good advertisement for the utility of MathOverflow to assist in the research work of a mathematician, I believe this would be such an example. This was especially significant for me at the time: as a first year Ph.D student, to have my question considered seriously by a mathematician of Richard Stanley's stature, was a major boost of morale. 

An example of an entirely different nature is given by the following question: 

The density of integers represented by a binary form

If you check the question, you will see that the accepted answer is by... me. In fact, when I asked this question I had just begun to realize that the asymptotic formula for the number of integers up to X which can be represented by an arbitrary binary form with integer coefficients and of degree 4 can be obtained. I didn't know that in the next few months I would collaborate with my advisor Cam Stewart to prove this asymptotic formula for ALL degrees; this was done in the following paper: On the representation of integers by binary forms.

This question, and the paper that followed, was another turning point for me. I gained the conviction that not only was I capable of asking good questions, but that I had the ability to answer them. 

Yet another example of the positive benefits for one's research program by engaging with MO, concerns the following question:

Polynomial representing all nonnegative integers

This is an example of MathOverflow having the ability to motivate and enthuse researchers well beyond its natural borders. Despite being a frequent MO user, this question was a bit before my time, having been posted in December 2009, about a year before I joined. As such I was not aware of this question for a long time. It was only in 2022 when a senior colleague mentioned this problem, by the name "Poonen's question", to me at a conference. He talked to me about this problem because he believed my earlier work on representation of integers by binary forms make me as qualified as anyone else to answer this question. 

Intrigued by the question, I would go on to work on it, in a collaborative effort, and achieve a modicum of success. This resulted in the joint paper: 

Quartic polynomials in two variables do not represent all non-negative integers

(to appear in the Journal of the European Mathematical Society).

My personal story is one where success in research can be found by engaging with MathOverflow in many forms. Perhaps this will compel you to think about how your research can benefit from increased participation at MathOverflow. 

Anatomy of a good MathOverflow question

In my experience of asking many questions and reading many more questions on MathOverflow, as well as interacting with the mathematical literature at large, I would say that good MathOverflow (and perhaps more generally, a mathematics question) question has the following common traits.

1) Providing sufficient background and motivation: mathematics is effectively an infinite expanse. There are many examples of infinite families of questions, for which very few (likely finitely many) are actually of interest to anyone. An interesting or worthwhile question necessarily has some additional background or story that makes it stand out from the rest. It is the job of the person who poses the question to inform the audience what it is that makes the question special.

2) Lends itself to a concrete answer: a good question should be answerable. To use a cliche example, the Riemann Hypothesis is a great conjecture because it can be answered definitively; either a proof is obtained giving a positive answer, or a single counterexample is enough to nullify it. 

This doesn't mean more nebulous questions are definitely bad. Mathematics is about exploration, and sometimes uncharted waters are murky and confusing. However, there should still be some way to answer the question, even if the answer is of the form "this type of question was considered in some paper in the past, and it is declared in that paper that such problems are out of reach". 

3) The question is self-contained: In principle, any reasonably well-trained person in mathematics should be able to understand what is being asked. This means that notation should be defined, and definitions provided whenever reasonable. This doesn't mean that every little thing should be defined: content that should be familiar to anyone who has done a typical bachelor's program in mathematics do not need to be defined. This includes things like sets, groups, continuity, Hilbert space, etc. 

Like all finite lists of rules, there are certainly going to be examples of good questions that do not fit the mold. As long as your question is driven by curiosity and good faith, I believe it will be well-received more likely than not, and who knows, you might get some good answers or interaction from the community!