Some fun fibers.
The first thing to do is to choose the materials. Which can be hard when you have so many choices! You will need a small piece of organza. A thread that will complement the organza, this will be for the wing edges. A small hoop, straw or other fine needle. Some fine wire, around 26 gauge should be flexible enough. You may also want a different thread for the veins on the wings. Can't forget some beads for the eyes and head. My dragonfly uses bugle beads for it's abdomen.
Once you have the organza you need to draw dragonfly wings on the fabric, These will be freestanding in the embroidery so space them apart for ease of working. I use a pen on the fabric, I know you shouldn't but it is hard to see light lines on organza. Just keep drawing until you have four wings. You will need them to also be right and left or opposites of two of the wings.
Dragonfly wings drawn on organza.
Shaping wire to dragonfly wing.
Once you have a good shape you can start embroidering. This is fine stitching so you will need only one strand of thread. Cut embroidery thread between 12 and 18 inches. It's better to have to add another thread then to have worn looking threads. I start by making a small stitch on the outline and then come up from the backside and split my stitch. This will take the place of a knot. Never use knots because they can pop to the front of the work. Place the wire on the fabric and roughly couch over the wire shaping it as you go. It always feels strange starting because the stitches are so tiny and the fabric so fragile. But keep with it because once the wire is couched to the fabric it gets easier. A lot of people take the end wires through the back of the fabric, I don't because then you can't see them and the thread gets tangled in the wire.
Once the wire is couched you will need to embroider over the wire. I find it easier to get a neat stitch if I slightly angle my stitches. When coming up angle from under the wire and up, then when stitching down angle slightly under the wire. It really helps to angle the stitches but if that is hard just stab stitch up and down. Be sure to work over the beginning thread neatly, These wings have to be as beautiful as possible and small details are essential. No messy backs! Repeat on other wings.
Couched wire being embroidered with one strand of turquoise thread.
Embroidered dragonfly wing.
Back of dragon fly wing.