All Recording Rarities fish locations in Dredge
The Dredge Recording Rarities pursuit is your classic 'sail to the four corners of the world to find the legends residing there' quest. In this case, those legends are four massive fish the travelling merchant asks you to catch, and for each one you show to her you'll get two Research Parts with which to unlock new gadgets for your boat. You can also sell each fish for around $250, meaning if you catch all four you'll be swimming in cash.
In order to complete Recording Rarities, you need to unlock explosives from Gale Cliffs, the Abyssal and Hadal rods from Stellar Basin, and you also need the Mangrove rod. I'd definitely recommend getting a better engine, too, since you'll be sailing to all four main areas on the map. Lastly, be aware that each of these fish take up lots of space in your cargo before you head out to wrangle them, but on the plus side, it doesn't matter if they rot or you sell them since you caught each and got the encylopedia entry to bring back to the merchant.
Now that's sorted, here's how to catch each Recording Rarities fish in Dredge.
How to catch the Oarfish
You can find the Oarfish in the Gale Cliffs region in square P3. Head to where you dredged up the Family Crest for the pursuit that unlocked explosives in the south-east of the island and you'll spot a waterfall. Sail straight through it to enter a hidden cave where you can catch the Oarfish. This one requires the Abyssal rod to catch. If you're having trouble with the aggressive one-eyed fish that patrols the area, remember that you can hide behind the rock as you enter the cliffs near the town to let it pass, or you can sail around the island and enter from the back.
How to catch the Gulper Eel
The Gulper Eel is located in Stellar Basin in square F4. You already will have visited the Research Platform when unlocking the Abyssal fishing rod, and the Gulper Eel is basically right next to it. Sail west a short way from it until you're above the underwater trench leading into the centre of Stellar Basin and you'll spy the big fish swimming around. For this one you need the Hadal rod, so make sure you've unlocked, purchased, and installed the Bottomless Lines upgrade. Also be wary of sailing over the centre of Stellar Basin unless you want those big tentacles to attack you.
How to catch the Goliath Tigerfish
The Goliath Tigerfish is found in the Twisted Strand in square E12. You'll need explosives for this one, so make sure you have some on hand, or buy some from the travelling merchant platform near the entrance. Simply sail south-west down the edge of Twisted Strand from the pontoon then north into the little gap. Almost immediately on your left you'll see a fallen tree you can blow up, leading into a little mangrove cul-de-sac where the Goliath Tigerfish is swimming. You'll need the Mangrove rod equipped to catch it.
How to catch the Coelacanth
You can find the Coelacanth in Devil's Spine in square Q12. Head to the southernmost island in the area to find the travelling merchants pontoon, then sail east around the island, being careful to avoid the glowing piranhas in the ruins nearby. Past the first set of ruins you'll spot a rock barrier you can blow up, leading into a little passage where the Coelacanth is swimming. You'll need the Abyssal rod to catch this one.
Take the fish back to the travelling merchant to complete the pursuit. Sadly, there's no extra reward for completing it other than what you'd usually get.