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
    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?