Decide whether you'd like to have clip on backs or studs, or use earring hooks.
 Here are some suggestions:


Model the clay into the shape of a dragonfly
Punch out the shape of a dragonfly in a circle of polymer clay; a twist on this is to punch out the shape in a different color and reinsert the different color into the punched out of the circle. This method requires a small punch in the shape of a dragonfly or some kind of improvised wire cutter in the shape of a dragonfly.
 One end should have a bulbous top for the head, while the rest should taper down finely to represent the back of the dragonfly. Take your time, as shaping it can be tricky but keep in mind that you will not get a delicate shape using polymer, so treat this as creating a resemblance or even a caricature of a dragonfly.
Shape the wings on each side of the body. Each side should have a pair of wing, a top wing and a base wing.
For eyes, either add two tiny balls of polymer clay to the ball head or simply poke two holes into the head.
Shape two small feelers at the top of the head.
Repeat to make the second earring.
Pierce a small hole into the upper body end of the dragonfly where the earring hook will be placed.
Bake according to the instructions accompanying the packaging.
Once baked, remove from the oven and allow to cool. When cooled, slip the earring hook through the hole and the dragonfly earrings are ready to wear.

Center the dragonfly punch or improvised cutting shape over the circle and punch out the dragonfly shape. If you are going to leave this as a hole, you can proceed to bake.
If adding a differently colored dragonfly center, punch out the shape from the other-colored polymer clay first. Then, gently push in the piece into the hole you've already punched out of the circle. You may find this a bit tricky and it's very important to do it carefully and slowly to avoid bleeding of the design colors into each other. Use a fine tool to help you, such as tweezers, rather than your fingers. Proceed to bake as instructed.
Repeat for the second earring.
Pierce a small hole into the upper body end of the dragonfly where the earring hook will be placed.
Bake according to the instructions accompanying the packaging.
Once baked, remove from the oven and allow to cool. When cooled, slip the earring hook through the hole and the dragonfly earrings are ready to wear.
Choose a suitable polymer clay.
Decide which approach to take to making the dragonfly earrings.
Assuming you've decided to model the clay into the shape of a dragonfly, do the following:


Begin with the body and make an elongated shape.
Assuming you've decided to make the punch-out version, do as follows:


Shape a neat small circle in the size you consider will be comfortable to wear.
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