Feebas can be found on B1F of Mt. Coronet, which is the room with the large underground lake:

You can only find Feebas in four random water tiles within this lake, and only by fishing. If you get the correct tile, Feebas will have a 50% chance of appearing. It doesn't matter which rod you use.
It's as bad as it sounds: you need to fish in every water tile until you eventually find Feebas. There are 528 fishable tiles, which means there is a 0.76% chance of any given tile having Feebas. To make matters even worse, the Feebas tiles roll over every day, so you should do your entire hunt on the same day.
There is one small technique that could help you get Feebas faster, which is mixing records. If you mix records, everybody's Feebas tiles will change to that of Player 1. So you can mix records with friends, then divvy up the lake between yourselves until you one of you finds Feebas. Then you can all go to that same tile and also find Feebas there.
It's most efficient to trigger an encounter on each tile once, and move on if it doesn't give you Feebas.
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