Remove the drywall compound, which forms the bulge to expose the head of the popped fastener. Older drywall installations were often fastened with nails that have a tendency to work loose, but screws can work loose as well.
Remove or reset the popped fastener. Use the claw back of a hammer to remove a nail. Remove a screw with a screwdriver or drill bit. You may also choose to drive the popped screw deeper, or to reset the nail with the hammer and a nail punch.
Install a new 1 3/8 inch drywall screw about 1 1/2 inches above or below the pop on the same stud.
Press the panel of drywall firmly against the stud as you install the new screw. The new screw should make a small dimple in the drywall without tearing the drywall paper.
Apply spackling paste or drywall compound with a wide-bladed putty knife to patch the wall. Vinyl spackling tends to shrink a bit more than traditional spackling, so you may need to come back and do a second application.
Sand down the spackling or drywall compound so it is even and smooth, blending into the rest of the wall.
Use a slightly damp rag to wipe up any sanding dust. Don't apply too much pressure or moisture to the wall -- you just want to pick up the loose dust.
Paint over the repair with primer, especially if you will not be priming the whole wall. Spackle and compound absorb paint, which will cause the area to look flat or dull. This is especially noticeable with satin or semi-gloss paints. Sealing the repair with primer first will ensure that it won't be visible after the room is painted.