Try placing a rubber mat or damp towel underneath the cutting board to prevent it from moving.
Make sure the knife you choose is very sharp. Getting a clean cut with a dull knife is hard. A dull knife will end up mangling a beautiful fillet.
When you’re first starting out, making a single vertical cut is easy and ensures you don’t pierce the skin. With practice, you can rotate the blade during this cut to transition to the next step. It allows you to skin the fillet with a single, efficient cut.
If you’re handy with a knife, turn the knife when you’re about ¾ of the way through the vertical cut from the last step. Cut diagonally down at a 45-degree angle toward the skin, then level out the knife in one fluid motion.
If you lose track of the knife’s positioning while cutting, lift the fillet again. The separated skin will fall forward, so you will be able to see how well you’re doing and adjust accordingly. Keep the knife between the skin and flesh. If you need to, reposition the knife to get a cleaner cut.
Basically, you don’t need to move the knife very much to cut a fillet. Fish meat is delicate and doesn’t take much force to cut through. Check your progress when necessary by flipping up the fillet until the severed skin falls off of it.
The key to skinning a fish is making a single, consistent cut with smooth knifework. It gets easier with practice. The leftover bits happen when you pierce the skin. Small patches of skin can be tough to remove without butchering the meat, so you’re often better off leaving them alone.
Some fish that have dark and light sides include flounder and fluke. The darker side is thicker, so skin it last. To stabilize a slippery cutting board, put a rubber mat or damp towel underneath it.
Removing the head is much easier on flatfish like sole and halibut. It’s simple to trim off with scissors or with a vertical cut. For other fish, such as trout and grouper, cutting through the gills with a knife will help you clean the fish. You could then slice horizontally from the cut to the tail to divide the fish into fillets.
Removing the fins is easier with flatfish. For round fish, consider cutting the fish into fillets or cooking it whole with the skin on.
If you cut through the tail, make a second cut near the first one. You may lose a little bit of meat, but it will help you remove the skin in one piece.
If the skin breaks at all, use the tip of a fillet knife to separate it from the meat. Then, continue peeling it off.
Another option is to cut your own fillets by slicing open the fish’s spine and belly. If you didn’t skin the fish before removing the fillets, do it afterward with a thin blade. You could also cook the fish with the skin on, then cut it off later. It’s easiest when you crisp the skin on the stove or grill.