In a great many detective stories, the detective will seize upon a single clue that, to him, does not fit, but—if we’re being honest—readily has an innocuous explanation. A partially drunk cup of tea, a single dumbbell, no bookmark in his book—it can be anything. The people around him will often scoff that there are many simple explanations, but he will persevere. Until that detail is adequately explained, we do not have the real solution.
This trope is often quite a lot of fun, but it can be a tricky rope to walk.
On the one hand, if the clue is too readily explained, the detective is not justified in clinging so tightly to it. On the other hand, if there is no reasonable explanation for the clue, then it becomes an obvious clue and no one is justified in scoffing at the detective. The main fun of this sort of clue consists in walking right on the edge—the detective being justified in clinging to it, but just barely.
A fun variant of this approach is for the detective to set the irksome clue aside and focus on other things, only to discover, in the end, what the meaning of the irksome clue was. This, I suspect, works best in detective stories that have the structure of a short story—clues, then turn the page for a denouement. (TV murder mysteries tend to have this structure, spread out over a little more time. This is why the detective so often figures out who did it after some almost meaningless clue which gets him thinking of things the right way—it’s the pretext used to give the audience some time to consider the clues presented up to this point and guess before the detective reveals it.) I suspect that the Poirot-style mystery where the clues are gathered in a confusing order and then Poirot sets everything straight in the accusing parlor would work to.
I tend to favor a more gradual process, where the reader is closer to the detective’s thoughts, so I haven’t used this sort of clue in my murder mysteries yet. I’ve also tended to stay away from solutions that hinge in a single piece of physical evidence. That’s not a policy, it’s just where I’ve gone. The critical piece of physical evidence is very tricky to pull off right—it’s very difficult to make it both conclusive and yet non-obvious.
The one clue which doesn’t fit is even harder to pull off in a novel, since it (more or less) has to be the key to looking at things from the correct perspective. From the correct perspective, things make sense and the evidence to get leaps to mind; in a novel it is hard to have this perspective but never have anyone consider it. It is, of course, doable—the key seems to be to have an alternative perspective which seems more plausible and almost makes sense, in which the detectives can labor until they come at the problem from the right angle. The problem with this approach is that though it does have the effect of startling the reader, I think that it diminishes the enjoyment of re-reading the story, since one knows that the detective is walking down useless paths. (To make them non-useless basically requires lucky discoveries, and luck is not what fun is made of in a detective story.)
It is interesting to consider how to use this sort of clue well. It can be a lot of fun, and I would enjoy working one into a novel that I write.