I have already caught the coelacanth, so I know how to catch one.
They are caught in the ocean. The coelacanth comes out all year when there is some sort of precipitation (snow or rain). They start to come out at sunset. I caught mine at about 8:00 p.m. and it was raining. The coelacanth's shadow is about the size of a sea bass's shadow (or bigger), so one fish you catch alot while trying to catch the coelacanth is a sea bass. If your planning to sell the coelacanth they sell at around 15,000 bells. Also they are known as the one of the rarest fish in the game.