Wikipedia lists the following interpretations of modal logic:
Alethic: fundamental conditions of possible worlds, causality, time-space parameters, and the action capacity of persons. Indicates the possibility, impossibility and necessity of actions, states of affairs, events, people, and qualities in the possible worlds.
Deontic modal operators influence the construction of possible worlds as proscriptive or prescriptive norms, i.e. they indicate what is prohibited, obligatory, or permitted.
Axiological modal operators transform the world's entities into values and disvalues as seen by a social group, a culture, or a historical period.
Epistemic modal operators reflect the level of knowledge, ignorance and belief in the possible world.
Doxastic modal operators express belief in statements.
Are you aware of more?
It seems that more interpretations may be constructed through the Categories of Being.