What is "folklore"?
First, because there appears to be some confusion in the comments, I think it is important to give some context. In contrast to other areas in academia, mathematical results can be proven to be true. Mathematics is not particularly empirical—we don't experiment or interview research subjects. A well-formed statement in mathematics is either true, or it is not (whether or not it can be proved one way or another question entirely... thanks, Gödel).
Published results in mathematics typically consist of one or more mathematical statements ("theorems"), along with whatever argumentation is required in order to prove that those statements are true (within whatever system of axioms and logical reasoning has been adopted for that publication). There are no "negative results", nor ethnographies, nor partial experimental results.
In this context, the culture of mathematics values novel results (i.e. statements which have never been proved before) or, barring that, novel methods (i.e. lines of argumentation which have never been used before). Proving an incremental result with standard techniques is typically not considered interesting, and will typically not merit publication.
In this setting, the term "folklore" has a fairly specific (if somewhat ambiguous or non-rigorous) meaning. The "folklore" of mathematics consists of results which are believed to be true, but which no one has bothered to fully flesh out or publish. Generally speaking, these are results that experts agree are true, and it is very possible that some of these experts have taken the time to walk through a proof, but haven't published, either because they seem too trivial or too tedious to bother with.
"Folklore" is not necessarily "common knowledge", nor is it necessarily "low-hanging fruit", though it can have connotations of either (or both). More typically, "folklore" consists of results which are assumed to be true, but which aren't particularly novel and/or appear to be amenable to standard techniques—these things typically don't rise to the level of "interesting enough to publish". These are results which are communicated primarily through word-of-mouth (e.g. from advisor to student, or within a particular working group), but which aren't published (for whatever reason).
For reference, there is a pretty good discussion of mathematical folklore on nLab (nLab is a kind of mathematical "wiki"—it is a collectively edited collection of articles which is meant to serve as a kind of "lab notebook" for mathematical reasearch—the article on "folklore" roughly reflects what (a select group of) mathematicians think about the topic).
Is it "ethical" to publish folklore?
Assuming that one has a proof of some folkloric result, there is nothing unethical about publishing that result. Indeed, I would argue that it is probably a Good Thing™ to publish these results, if for no other reason to have it written down for the sake of peer review (both in the context of having a committee of experts vet the result, and in the context of having more eyes in the community to spot bugs).
Is is a Good Idea™ to publish folklore?
*shrugs*
Ideally, if an old result is being used to prove a new result, then a proof of the old result should either be cited (if it exists in some other paper or text) or explicitly given as part of the exposition leading to the proof of the new result. If part of proving a new result includes allusions to the "folklore" of a field, then note that it is a folkloric result, give a proof, and move on.
It is also, in my opinion, reasonable to put forward a paper which seeks to do nothing more than prove a result from the folklore, though there are things to consider when going down this path.
From a CV / career standpoint, it very much depends on how advanced the author is:
For a student, I would think that attempting to publish a folkloric result can only be a good thing. Because of a perceived lack of novelty, it is unlikely that such a result will go into a "good" journal, but students (undergraduates and graduate students alike) typically have very few publications, and any publication is going to look good.
There is also nothing wrong with a late-career researcher publishing such results. They have nothing to lose (e.g. they have tenure and are not looking for further career advancement), and are potentially adding to the published record. Why not?
However, for an early career researcher, one should carefully consider the opportunity cost in pursuing folkloric results. Is it a good use of your time to write a paper that is probably only ever going to get into a C+/B- journal? or would it be better to work on something that is going to go into a better journal? There is also a chance that publishing this kind of result will just look like CV padding, but this is going to depend very much on whoever is reading over that CV.
From the point of view of recording human knowledge, I think that there are many good reasons to go ahead and publish folkloric results:
These results are sometimes wrong. If someone manages to prove that something in the folklore is wrong, then it needs to be recorded, particularly if many other results depend on the incorrect folkloric assumptions. Moreover, I would imagine that someone who can disprove a result in the folklore would have a huge leg up on the job market.
Even if the results are not wrong, they may be very niche. An outsider, not immersed in the culture of a particular sub-sub-sub-sub-field might not know the folklore, and might waste time on something which is already "known" (imagine the physician who reinvents calculus, but with something known only to a small number of specialists). It is helpful for the dissemination of knowledge to publish these things, even if it is not a huge career boost.
Many specialist fields in mathematics are opaque to students. It is helpful to publish the folklore if for no other reason than to make the barrier to entry that much lower. Publishing the results means that students can learn the material, which makes it easier for new researchers to enter the field.
Overall, my preference is that these results get proper peer review and publication. They are likely not to go into particularly prestigious journals, but I think that it is valuable to make folklore explicit—a "shadow repository" of folkloric knowledge is kind of antithetical to the ideals of academia—it makes knowledge less accessible and circumvents peer review. I am on the side of "publish".
