Purchase a wall anchor kit. These kits typically come with wall anchors in different sizes, from plastic to metal, with screws that work with each size.
Drill a hole into the plaster with a masonry bit approximately the same size as the wall anchor and screw you plan to use. Do not use an oversize bit, as it does not make a hole small enough to secure the anchor.
Place the end of the wall anchor into the hole and carefully tap it in with a hammer while holding the anchor as straight as possible. Do not tap too hard or force the anchor, especially if using a plastic one, as this can cause it to bend and become useless.
Slowly screw the screw into the wall anchor using a drill with a bit that fits the screw; for a flat-head screw, use a flat-head bit, and likewise for a Phillips-head screw. Leave a small amount of the screw sticking out so it holds up the picture or wall art.
Drill a hole into the wall at an angle, approximately at the same degree the nail goes in through the picture hanger. Make sure the hole you drill is slightly smaller than the nail, and use a masonry bit to ensure you can get through the plaster with ease.
Place the nail into the picture hanger, pushing it through both holes.
Set the end of the nail on the pre-drilled hole and push until the back of the hanger is flat against the wall and level.
Gently tap the head of the nail into the hole with a hammer, working slowly to make sure you do not crack the plaster. Hammer until the head of the nail is flat against the hanger and the hanger is secured to the wall.