Is there a sensible classification of the properties of structures with a given signature $\sigma$, e.g. graphs with $\sigma = \lbrace R \rbrace$?

For example like this:

properties defined by first-order sentences over $\sigma$ (with or without specific syntactical properties)

properties defined by monadic second-order sentences over $\sigma$ (with or without specific syntactical properties)

properties stating the existence of another structure (of the same kind or another) and a specific

~~mapping~~relation to itproperties stating that a given structure invariant has a specific value

properties stating that a given structure invariant has a specific property

How can this list be expanded, can it be completed or can it never be exhaustive?

downfrom your number (1) -- geometric logic, essentially algebraic logic (finite-limit theories), algebraic logic. There are many variations. There is undoubtedly no end to such classifications, both up and down from first order logic. – SixWingedSeraph Jan 8 '10 at 15:13