Although Doug could be right (regards to condensation and dust/soot from forced air heat (heating oil fuel or natural gas), more likely you have a two-fold problem, namely condensation AND mold/mildew.
Cool air, or just the cold from the outside wall and poor insulation can allow for the cold to be transmitted via the framing elements of these walls. Condensation being "wicked" through the wood framing, through the drywall/drywall; and, as most dry wall is "taped" with paper tape or fiberglass with an adhesive, then "mudded" smooth (dry wall or "spackle compound" the seams and over the nail heads (in pairs) or screws, and that same moisture is being "wicked" via the nails/screws and they are "popping".
The black streaks may be dirt/dust/soot colecting on the condensation/cold areas or it could be mildew or MOLD growing on these condensation (wet)/cold spots, or a combination of two or three of these elements.
Unfortunately that you have these "black streaks" on walls that carry from exterior walls, and on exterior walls, but not interior walls, indicates that you may have a MOLD and/or mildew problem, and that these exterior walls are not properly waterproofed (perhaps just damp proofed), nor insulated properly.
Unfortunately prime temperature environment for various mold and mildew colonies is oftentimes between 35 and 110 F.
There are some types of mold (black, but some times other colors) that are extremely toxic to humans and pets, although there are just as many that are mostly benign.
IF you have toxic mold, or a less dangerous mold or mildew problem, you may have a situation where the lumber framing is contaminated and has been growing for a while and is just now reaching through the paper edging on your "dry wall" or "drywall". If that is the case, surface cleaning of the drywall situation will not stop the problem.
I would recommend that you investigate the following:
1. If its "soot" it won't brush off easily, and would need warm water with a surfactant (like detergent) to remove most easily.
2. If its mold/mildew to remove, might wipe away with warm water (then reappear a few hours/days/weeks later) or require a bleach and water solution with a few drops of dish liquid (like dawn - but NOT antibacterial) to remove. 2 ounces regular clorox bleach and 19 ounces water plus 2 drops Dawn, Palmolive, or other liquid dish detergent (not dishWASHER detergent).
Until you solve the question of the origin of the problem (COLD/condensation) (air leak, insulation, waterproofed foundation, etc.) then correct and repair accordingly, this will be a continuing problem each cool weather season, and if you have a mold/mildew problem, you will need to take corrective action. Before dealing with a suspected mold/mildew contamination situation, would advise consulting with an environmental engineer expert and getting samples tested at an accredited laboratory (check with your local EPA for a list of accedited labs), and if you have toxic mold, hire professionals to correct the problem, as cross contamination is quite easy and the toxic molds are major health hazards.