At the time of writing the Common Carp is only found in Fisherman's Rest (part of Fawnmirror Lake).
1. Drive your boat to the far right of the level so that it stops automatically. You'll now see 2 piers/docks. The larger of the two is behind your boat and has a run down shack/boat house/shed on it.
2. Move to the left about 1 screen width so your boat's engine is almost level with the end of the dock and then turn your boat to face to the right (so you can fish to the left).
3. Pet your dog/cat because they deserve it.
4. Change the time of day to Dawn.
5. Equip the Deep Crankbait lure and the Horizon Caster rod and tap
to cast the shortest possible distance (the rod and lure I mention is purely to help you locate the right trench for the fish, catch the thing with whatever you want). The Legendary Common Carp will be at the bottom of the V-shaped trench that your lure falls into.