Is it meaningful to publish results that are not interesting? (I'm by no means in the situation described below, I'm just hypothesizing/daydreaming.)
Say that a professional mathematician thinks that he can get some interesting results by applying a certain method he found. Being a seasoned mathematician, he is confident in his attempt, only to find out that the results yielded by the method were trivial. Is it still meaningful to publish the result, as to tell other mathematicians that come up with the same method "Hey, I've already checked this cave, its empty!"? In particular, is this frequently done?
 A: This paper is great: How not to prove the Poincare Conjecture by John Stallings (now deceased unfortunately.) A great mathematician with a great sense of humor.
Bibliographic info:
Stallings, John,
How not to prove the Poincaré conjecture. Topology Seminar, Wisconsin, 1965, 83–88,
Ann. of Math. Stud., 60, Princeton Univ. Press, Princeton, NJ, 1966. 
A: Taking the viewpoint that "attempts" are "experiments" which may or may not yield information, a "failure" may yield considerable information even while not achieving a decisive conclusion... and, therefore, be interesting.
Oppositely, some experiments seem fairly pointless "in advance", and/or prove unsurprisingly boring and uninformative, ... so are not interesting.
A possibly useful distinction is that a relatively expert mathematician may be able to relatively easily anticipate that a given experiment will yield no (interesting) information, while a relatively inexperienced person might not see the pointlessness of going down a particular road. At the same time, an inexperienced person may themselves acquire to-them-valuable information by performing to-expert-pointless experiments, to see first-hand (and viscerally?) how things work. Yet, as privately informative as this might be, it would fail by the stronger criterion of "informative to experienced experts".
Altogether hard-to-say, subjective.
Also, "it depends" on what one means by "publication": trying to get something "peer-reviewed" to score professional points is quite different from literal publication, e.g., on the internet, to inform any interested parties.
A: In most cases, no, negative results are very hard to publish. Stallings's situation was an exception; recall he had published major work on this subject. 
This is a general nuisance in science, not only in math. If you look at your favorite math journals they rarely if ever will they have papers showing a possible method doesn't work. 
