The lack of hard parts in soft-bodied organisms makes them extremely rare in the fossil record. It takes very special conditions for them to be preserved where they are buried rapidly in areas devoid of oxygen and scavengers.
The scientific definition of fish is complex and more than a little confusing. So here is a much simplified definition. Fish are: chordates, they have a backbone or notochord (cartilaginous support), live entirely in water, have apparatus to extract oxygen from water (gills), appendages without digits, fins, and are generally ectothermic. There are exceptions to these characteristics, for example some fish can live for long periods of time out of water and some fish, like tuna and large sharks have body temperatures well above the temperature of the surrounding water, but for the most part these characteristics describe fish. Fish can have jaws or be jawless, have scales or skin, drink water or not. A number of organisms we call fish actually are not fish at all. Crayfish are not fish, they do not have a backbone, and their appendages have digits, in this case claws. Likewise shellfish are not fish, this is a marketing term to refer to organisms from the water that have shells. There are about 32,000 species of fish that live in both freshwater and saltwater and from the high mountains lakes to the intertidal zone to 23,000 feet deep in the ocean abyssal plains.
Fossil fish can be found on every continent, including Antarctica. There are a number of significant fossil sites around the world.
* The Green River Formation, Wyoming produces large quantities of fossil fish including rays. It is also known for its amazing preservation which results is some of the most stunning fish fossils.
* Solnhofen Limestone, Germany. This site has produced some of the most amazing fossils ever discovered, including the famous Archeopteryx fossil. It also produces fish and other marine animal fossils.
* Some of the most exquisitely preserved, Cretaceous fish in the world come from the limestone deposits near Byblos, Lebanon. The discovery of amazingly preserved fossil fish dates back many centuries. In fact they were first mentioned in writing by Herodotus, over 450 years before the birth of Christ.
* The Crato Formation of Brazil has yielded twenty five species of early Cretaceous fish, many with their stomach contents in tact.
* The Gogo Formation, Australia has produced fifty species of fish, many with soft body preservation of anatomical features as small as nerves.
These are only a few of the most famous fish fossil sites.