A habitual issue nearly as iconic as the brand itself is a Gibson Les Paul with a broken headstock. This is the classic hinged break where the headstock is still held to the neck by the face overlay. These are the easiest of head breaks to repair since the headstock remains aligned during the gluing and clamping process. Just inspect the joint for anything that will prevent a tight joint, make the necessary adjustments, apply glue (hot hide glue is my preference) and clamp.
My 2 cents on why you see these breaks most often on Les Pauls versus Gibson acoustics or even SG's is that the Les Paul bodies are so damn heavy (SG's tend to break at the heel). When the guitar falls with such a huge counterweight it easily leverages against the short grain in the headstock and pops them open like a can of beer. Again, just my 2 cents.