Don't throw away your dull scissors. Sharpen them yourself with any one of six common household items you likely already have. The technique, which sharpens the scissors by smoothing out the microscopic pits in the edges of the blades, is basically the same no matter which material you choose to use.
Open the scissors wide and put one blade on the inside of a glass jar and one on the outside. Then close the scissors with the back pivot point, not the pointed tip, grasping the jar rim. With constant and even pressure, squeeze the scissors together as if you're trying to cut the glass. This motion drags the blades across the glass, from the back of the blades to the tip. Repeat until the blades have a smooth, clean edge.
Open the scissors wide and close them around a screwdriver's metal shaft. Again, with constant and even pressure, try to cut the screwdriver, dragging the blades along the metal shaft, from the pivot point in the back to the tip of the blades. Repeat until the scissors are sharp again.
Fold a piece of medium-grit sandpaper or emery paper -- a kind of sandpaper used just for metals -- in half, rough side outside, so both blades are sharpened at the same time. Cut the paper into strips. Repeat until the scissors are sharp.
Although the technique is the same, cutting into aluminum foil or steel wool requires a more precise effort to make contact with the entire edge of the blades. Again, cut foil or steel wool, making contact from the back pivot point to the tip of the blades. Repeat until you feel the scissors cutting smoothly.