In the .blend I am working, some day, in edit mode, I started to see the edges like this. (The bottom right box is the same mesh appended to a new session, like I want to see them again!).
I tried with no good result (all in edit mode):
3D View - Properties - Mesh Display - Overlays - All options
3D View - Display - All options
Object Panel - Display - All options
Object Panel - Display - Type
I compared the User Preferences with a new session preferences, and all the options are the same.
Blender 2.53.1 r31615M, from GraphicAll. Ubuntu 10.04.