This is odd and more common than you’d expect. “Shadow Staining” the sooty, dusty stains along the walls or ceiling that track along the wall studs or ceiling joists.
Basically what happens is the wall studs or ceiling joists are cooler than the interior air. The most likely culprit? Poor, improper or a lack of insulation. This temperature variance causes a very slight film of condensation to develop on the surface. Now, any dust or soot that may be in the air will eventually land and be caught by the moisture.
Sometimes you will be able to see every stud in the wall.
WHAT SHOULD WE DO?
I would clean it off. in most cases it will clean off with soap or bleach and water. Then paint over it with an oil based paint designed to cover stains like Binz or Kilz or Enamelac.
Keep in mind, this is a thermal transfer mechanism so unless you fix the insulation issue there is a good probability it will come back.