Nothing's more frustrating than getting a hole in your stockings. If it's just a tiny hole and the stockings are just beginning to develop a run, you only need to dab a little clear nail polish on the area and let it dry to fix your problem. If it's a larger hole, however, more elaborate measures must be taken. This is basically a sewing project, and with patience and a deft hand, you'll have your stockings fixed in no time.
Insert the needle threader into the hole on the top of the needle, push the thread through the needle threader, and pull it through the hole in the needle. Nylon thread can be difficult to work with, since it's so stretchy, but that stretchiness is exactly what you need for your stockings.
Draw the needle and thread carefully through one side of the hole and then the other. Repeat in the other direction, pulling the thread taut but not too tight. You don't want the delicate fabric to bunch or twist. Your goal is to make the stitching as flat and invisible as possible.
Finish off by pushing the needle into the fabric at the end of the hole and tying off a tiny knot on the inside of your stockings. Cut the thread with your scissors, leaving a quarter-inch length of thread.
Brush a small amount of clear nail polish along the seam to prevent any runs from developing from the area where you've sewn. Let the polish dry completely before wearing the stockings.