It would be neat if all actions could be sorted into three jointly exhaustive classes: the permissible, the impermissible, and the obligatory. These deontic modes would then be analogous to the alethic modes of possibility, impossibility, and necessity. Intuitively, the permissible is the morally possible, that which we may do; the impermissible is the morally impossible, that which we may not do; and the obligatory is the morally necessary, that which we must do.
Pursuing the analogy, we note that the following two alethic modal principles each has a deontic analog, where 'p' ranges over propositions and 'A' over actions:
1A. If p is necessary, then p is possible, but not conversely.
1D. If A is obligatory, then A is permissible, though not conversely.
2A. P is impossible iff ~p is necessary.
2D. A is impermissible iff ~A is obligatory. (e.g. Sexually molesting children is impermissible iff not sexually molesting children is obligatory.)
But of course the following ab esse ad posse principle has no deontic analog:
3A. If p is true, then it is possible that p is true.
3D. If A is done, then A is permissible.
This is not surprising since the deontic is a species of the normative.
Now let's try to characterize (not define) the deontic concepts of the permissible, the impermissible, and the obligatory in terms of the axiological concepts of good and bad:
4. If A is impermissible, then A is bad to do and good to leave undone.
5. If A is obligatory, then A is good to do and bad to leave undone.
6. If A is permissible then A is axiologically indifferent: neither bad to do and good to leave undone, nor good to do and bad to leave undone.
These characterizations seems to leave conceptual room for two other sorts of actions, actions that are good to do but NOT bad to leave undone, and actions that are bad to do but NOT good to leave undone. The former are supererogatory in a wide sense of the term, and the latter are suberogatory.
For example, paying one's bill in a restaurant is (legally and morally) obligatory: it is good to pay the bill and bad to leave without paying. But tipping is (legally, though perhaps not morally) supererogatory: it is good to tip but not legally required, and in this sense not bad to leave no tip. If this is unclear, consider tipping in excess of 20%. Surely this would be supererogatory: good to do (ceteris paribus) but not bad to leave undone. One could plausibly argue that restaurant patrons are morally obligated to tip in the amount of 15-20% of the tab given that waiters and waitresses work for minimum wage and rely on tips for a livable wage; but surely no one is obligated to leave a 50% tip, say. And yet it would be a very nice thing to do for that pretty waitress working her way through school. It would be good to do but not bad to leave undone.
Do you balk at calling such trifling actions supererogatory because they are not grand or heroic like throwing oneself on a grenade to save one's buddies? Well then, I will accommodate your point by distinguishing between wide and narrow senses of 'supererogation.' Or I might just give you the word as long as you grant me a category of actions an example of which I have just provided.
Now for some examples of suberogation. A supererogatory act is a good act that is in excess of what is morally or legally required. A suberogatory act is then one that is bad but not so bad as to be impermissible. The following might be examples: talking too much, public displays of affection, breaking wind in public, picking one's nose in front of others. The practitioner of public rhinotillexis does nothing legally or morally impermissible, but it is something bad, and not just aesthetically. I want to say that it is a morally bad action that not impermissible. So I say it is suberogatory.
More needs to be said. I am not sure that in the end the sub- and supererogatory don't collapse into the permissible thus obviating the need for two further categories of actions. Further reading: the SEP article, Supererogation.