Clipping the wings of your chicken may be needed if she is a good flyer and a 6ft fence doesn’t deter her from exploring the neighborhood. You should not clip her wings as a workaround for fixing the fence or if she doesn’t show the desire to fly away.
Clipping her wings actually consists of cutting part of the primary flying feathers. There is no surgery involved. The primary feathers have no blood in them. It’s a painless procedure similar to clipping your nails.
This video shows how to identify the feathers to cut and how to cut them. It’s helpful to have another person to hold the chicken while you inspect and then cut the feathers. The key is to take your time to identify the feathers to cut, and have a very sharp pair of scissors.
Before starting the procedure, have some gauze pads and styptic powder (alternatively corn starch would do) nearby. In case you clip another feather by accident and it bleeds, stop the bleeding immediately with the styptic powder.
Always clip both wings. The goal is to prevent her from taking off, not to have her fly in circle. If you clip just one wing, she can lose her balance, injure herself, and be confused when trying to fly.
Note that the feathers will regrow after her next molt, so you may have to clip her wings every year if she still shows the desire to fly away. But in my experience, they don’t try flying after having their wings clipped once.