The ability of modal assertions to define natural and interesting
classes of frames (or digraphs) is indeed intensely studied and
constitutes one of the principal perpsectives of the subject,
pervasive in all the literature and textbooks. Indeed, I heard
Blackburn assert at a conference talk last fall that one
should think about modal assertions *mainly* as a way of
describing certain classes of graphs.

Any of the standard reference texts on modal logic will tell you
that:

 - the modal theory S5 characterizes the equivalence relations;
 - the modal theory S4.3 characterizes the linear pre-orders; 
 - the modal theory S4.2 characterizes the directed partial pre-orders;
 - the modal theory S4 characterizes the partial pre-orders;
 - And so on.

There are numerous instances of this phenomenon for various
logics, and modal logicians are particularly interested in logics
with the finite frame property, which are those definable as
arising from a class of finite frames.

In some of my recent work, [Structural connections between a forcing class and its modal logic](http://jdh.hamkins.org/a-forcing-class-and-its-modal-logic/), for example, we have been looking
at all those logics and also what we call S4.tBA,
topless-Boolean-algebra logic, which is characterized as the
assertions true in every finite topless pre-Boolean algebra (a
finite pre-Boolean algebra whose maximal cluster has been removed). We keep being pushed toward the idea that this may be the modal logic of class forcing, and also of c.c.c. forcing. The
connection between the modal assertions and the nature of the
frames is exploited throughout the work.