You need to store the fish off the ship - so if your ship decays you don't lose them.
You don't need any "a fish", and the non-named crabs and lobsters are only good for cleanup points or library donation.
For fish:
If you caught it in shallow water, and it's named - you'll need it for quests
If you caught it in deep water, it's named, and it stacks - you'll need it for quests
If you caught it in a dungeon, it's named, and it stacks - you'll need it for quests
Named stuff that doesn't stack - These are mostly odd recolors of the basic fish, though there's a marlin form that's not. They have non-standard (not 10 stone) weights, and above 20 stones will display weight, fisher name and date.
The Mud Puppy and the Red Herring from the Underworld are for quests, and can be mounted. Otherwise they can be cut up for fish steaks to donate.
Some of the others can be mounted
The rest are used in pie recipes that act as buffs
There are
12 shallow water fish for quests
18 deep water fish for quests
12 dungeon fish for quests
6 crabs for quests
6 lobsters for quests