Some background from the page on Deontic Logic:
As discussed elsewhere, it seems difficult to represent ideas like "A commits/requires you to B" using just ⊃ and SDL, no matter which scope we use for the O operator. Some theorists respond by addding a different, non-truth-functional (and sometimes non-monotonic) conditional; others respond by adding a dyadic O operator, akin to the conditional probability operator that Bayesians use. Of course, many theorists also advocate using a deontic logic weaker than SDL; but probably one of these other devices will be needed, too.
Another complaint against SDL, also mentioned elsewhere, is that it can't express the idea that if you violate obligation X, "the least you can do" is to fulfill a weaker obligation Y.
Some puzzles confronting SDL have to do with counter-to-duty obligations; that is, obligations having to do with what you should do if you violate some other obligation. Some things meeting that description might be weaker, "the least you can do" obligations as mentioned above. But let's restrict our attention to obligations that come into effect only in the cases where you've violated some other obligation; in fact, they might describe things you're obligated not to do if you've fulfilled the other obligations.
A classic example is Chisholm's Puzzle (1963). In plain English: You ought to visit your grandmother, though in fact you won't do so. If you do visit, you ought to tell her in advance that you're coming. If you don't visit, you ought to refrain from telling her in advance that you're coming. (This is the counter-to-duty conditional, because it applies to you in the case that you violate your duty to visit.) The difficulty here is to formulate all of these claims using the expressive resources of SDL in such a way as to respect our intuition that all the claims may be true, and also our intuition that all the claims are independent. That is, no one or two of these obligations, plus the fact that you won't visit, entail the third. This doesn't seem possible to achieve.
If you intepret the "If you do visit..." obligation narrowly, as you do visit ⊃ O(you promise a visit), then it is already entailed by the fact that you won't visit. So --- of the resources we're allowing ourselves --- that had better be interpreted widely, as O(you visit ⊃ you promise). That leaves open whether the "If you don't visit..." obligation should be interpreted widely or narrowly. If it's interpreted widely, as O(you don't visit ⊃ you don't promise a visit), it is already entailed in SDL by the fact that O(you visit). So it had better be interpreted narrowly, as you don't visit ⊃ O(you don't promise). But then in that case our scenario is inconsistent, as given axiom K, O(you visit) and O(you visit ⊃ you promise) entail O(you promise). And in SDL, it cannot be that O(you promise), you don't visit ⊃ O(you don't promise), and you don't visit.
Some take this puzzle to motivate the abandonment of axiom K. You see suggestions like: perhaps O(φ ⊃ ψ) ∧ Oφ is not enough to entail Oψ; you need in addition the (non-modal) fact that φ. Or perhaps O(φ ⊃ ψ) ∧ Oφ entails only O(φ ∧ ψ), and that does not suffice for Oφ (thus rejecting M, a theorem of K and thus of SDL). The puzzle may become more tractable if we move to a dyadic deontic operator, where O(you promise | you visit); but by itself this doesn't resolve all the difficulties. On its standard handling, that together with O(you visit) still entail O(you promise), and that seems hard to reconcile with our judgment that, since you're in fact not going to visit, O(you don't promise).
A puzzle similar to Chisholm's (and also the Good Samaritan and Robber Paradoxes) is Forrester's "Gentle Murderer": O(not kill), but if you do kill you ought to kill gently, and in fact you will kill. Does it follow that O(kill gently)? Then in SDL it would follow that O(kill), which is not only counter-intuitive but according to SDL inconsistent with our starting assumptions.