Question

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:

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

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

3. properties stating the existence of another structure (of the same kind or another) and a specific mapping relation to it

4. properties stating that a given structure invariant has a specific value

5. 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?