The process for making a stage prop stand up on its own depends on the stage prop, but for any upright flat piece, such as a wall flat or wood or cardboard cutout, a standard device known as a "stage jack" works well. Once you have built a few simple stage jacks, you can use them for all kinds of props and scenery items. Larger items may require two or more stage jacks to stabilize them.
Cut a piece of 2-by-4 that is 2/3 the height of your prop and another that is 1/4 of its height. Stand the long piece upright and lay the short piece on the ground, butting up against it so from the side it forms a tall, thin "L" shape. Drill two holes through the bottom of the tall piece, into the short piece, and secure them together with 3-inch screws.
Hold the "L" upright and have someone else hold another 2-by-4 diagonally next to the "L" so it crosses the top and bottom of the "L," right at each end, forming a triangle. Draw a line at both points where the long piece crosses the "L."
Set your miter saw to the angle of each pencil line and cut the wood along those lines. Place the cut piece on the "L" to form a triangle and secure it with two screws at each end, pre-drilling to prevent splitting the wood.
Hold another piece of wood across the tall piece and the angled piece (hypotenuse of the triangle), about halfway between the base and the top, and again mark where it crosses both pieces inside. One line should be straight and the other will be angled. Cut both of these lines and secure this piece inside the triangle as a brace. This completes the jack.
Place the jack behind your prop and screw the prop to it with 1-inch screws, if possible. If the prop is three-dimensional and won't attach this way, you can also tie it to the jack or use any other method that securely attaches the prop without destroying its look.
Set the prop onstage and either screw the base of the jack to the stage floor or, for lighter props, place sandbags on the base to weigh it down and keep it from tipping.
Mark the jack clearly with pieces of glow tape, available at theater supply stores, to ensure actors and technicians don't trip over it during blackouts.