In my experience barbecue grease is a splatter pattern, rather than lines. The brown spots on the sills suggest spider s**t - could the lines be spider web residue? If you squash a spider web against the glass, the adhesive sticks along the line of the web filament and it can be a pain to get off.
I also had a row of jobs alonside a railway line and their sills were covered with brown spots from the minute particles of steel from the train wheels - these rusted and left immovable brown stains in the plastic.