What makes a fish kosher? It says in the Torah (Vayikra 11:9) that kosher fish must have “snapir v’ kaskeses” fins and scales. However all fish that have scales have fins, so in practice all one needs to determine that a fish is kosher is that it has scales.
Nevertheless not all fish that have scales are kosher, in order to be kosher the scales need to be easily removable and not so embedded in the skin that if one were to try to remove it they would have to remove skin as well. When purchasing fish if it is not being sold under Rabbinical supervision the rule is that the fish must still have some of the scales still on it as to determine that it is indeed a kosher fish.
This is a list of many popular kosher fish: Perch, Tilapia, Salmon, Halibut, Sea Bass, Trout, Carp, Whitefish, Pike, Tuna, Red Snapper, Yellow Tail, Mackerel, Cod, Flounder, there are many more as well.
BYOK (bring your own knife) while bringing your own knife to the fish store can be a good idea it is not necessary, if you inspect the knife that is used to cut your fish and it is indeed perfectly clean and the fish is cut on clean paper covering the cutting board it is permissible.