From the Super Mario Wiki
Super Mario series
Super Mario World
Super Mario World marks the first appearance of Fishin' Lakitu. Only one appears in this game, in Forest of Illusion 4. In this game, the Fishin' Lakitu dangles a 1-Up Mushroom from a fishing pole; should Mario grab the mushroom, it becomes a normal Lakitu and throws Spinies at Mario. Mario can defeat it the same way he can a normal Lakitu. It will respawn as a normal Lakitu from there on in.
This game also introduces the similar Fishing Boos.
Super Mario 64 / Super Mario 64 DS
No Fishin' Lakitus appear in Super Mario 64 and Super Mario 64 DS outright; however, the Lakitu Bros., who have fishing poles with cameras on them, appear in the game as cameramen and exist mostly to explain how the camera works.
Super Mario Advance 4: Super Mario Bros. 3
In Super Mario Advance 4: Super Mario Bros. 3, whenever the player wants to scan an e-Reader card, a Fishin' Lakitu will appear to whisk Mario away from wherever on the map he's on. This Lakitu has a slightly different appearance from the enemy Lakitus found in levels.
He goes unused in PAL versions due to all e-Reader content being dummied out. He also goes unused in Virtual Console releases due to all e-Reader content being unlocked by default.
Super Mario Odyssey
Fishin' Lakitus (referred to simply as "Lakitus") return in Super Mario Odyssey as NPCs instead of enemies. They appear in Sand Kingdom, Lake Kingdom, Snow Kingdom, and Bowser's Kingdom, and generally do not appear until all the main story Power Moons have been collected for a kingdom (most notably, the Sand Kingdom's Lakitu is frozen). Mario can use Cappy to capture the Lakitu and fish for the biggest Cheep Cheep in the water in order to get a Power Moon. In the Sand Kingdom, Mario must also fish Captain Toad out of sand, while in Bowser's Kingdom, he must fish for Poochy in purple sludge. If an enemy gets too close to the Lakitu, he will complain that he can't fish under these conditions, regardless whether Mario has captured the enemy in question or not.
A Fishin' Lakitu appeared in Mario Kart-themed issues of Super Mario-Kun, generally serving the same role it does in those games. This one seems to be wilder than usual; most notably, it bonks Mario with the starting light at one point.
Mario Kart series
A single, unnamed Lakitu appears throughout the Mario Kart series, billed in Super Mario Kart's instruction manual as the owner of all the tracks. He uses his fishing pole to hold the starting lights (except in Mario Kart DS) and fish out racers who fall off the track. The action of falling off a track and being fished out incurs a considerable loss of time and often position, and in Super Mario Kart, Mario Kart: Super Circuit, Mario Kart 7, and Mario Kart 8, he also charges up to three coins for fishing out a racer; however, if a racer does not have enough coins, he fishes out the racer anyway.
Starting with Mario Kart: Double Dash!!, he uses his pole to show the lap count as well as a U-turn sign when a driver is going the wrong way. In Mario Kart 8, after about five seconds, he will also flip a driver so that they go in the right direction again.
Mario Kart: Double Dash!! has an intro gag regarding the Fishin' Lakitu: he normally appears holding the starting light, but may sometimes appear with a Cheep Cheep, which may be so big it falls off the hook.
Yoshi's Island series
Super Mario World 2: Yoshi's Island
In Super Mario World 2: Yoshi's Island and its remake, they are present only in Hookbill The Koopa's Castle, where they attempt to steal Baby Mario from Yoshi's back with a baitless fishing pole ending in a hook. If they snatch Baby Mario, the Countdown Timer will start. However, these Fishin' Lakitus fly away from Yoshi with their catch, making it harder to save him.
Yoshi's New Island
Fishin' Lakitu returns in Yoshi's New Island, unchanged from their appearance and abilities in Super Mario World 2: Yoshi's Island.
Super Mario RPG: Legend of the Seven Stars
Mario Tennis series
In most Mario Tennis games, a Lakitu sits on one side of the net and uses his fishing pole to fish balls. In the Nintendo 64 Mario Tennis, he's only seen doing this when a player faults. In Mario's Tennis, he shows the current score using his fishing pole, while in Mario Power Tennis, he uses the pole to show starting lights (recycled from Mario Kart: Double Dash!!), the score, and time left in various minigames.
Mario Party series
Lakitu occasionally uses a fishing pole in the Mario Party series. This is seen in Lakitu's Luck in Mario Party-e and Test for the Best in Mario Party 8, as well as in Toad Scramble mode of Mario Party: Star Rush. Fishin' Lakitus appear in Super Mario Party as non-playable characters.
Mario Golf series
In the Mario Golf series, if a player hits into a water or tree hazard, a single, unnamed Lakitu will be seen with his fishing pole, presumably to fish the ball out. This was not seen in the GBC Mario Golf in favor of an angry Blooper shaking a fist at the player, nor was it seen in Mario Golf Advance Tour.
A Fishin' Lakitu also appears at the start of Speed Golf, holding starting lights (usually recycled from a Mario Kart game) on the fishing pole.
Mario & Luigi series
Fishin' Lakitus appear in the first three games in the series (as well as the first and third games' remakes), acting as level-up assistants. Whenever the player levels up, a Fishin' Lakitu appears with a selecting device that allows the player choose which stat they wish to add bonus points to (on top of the normal points gained while leveling up). Once selected, the device then flips over and becomes a roulette that determines how many points that stat will earn, from one to three. Fishin' Lakitus also appear in the final battle with Fawful in Mario & Luigi: Superstar Saga, holding up spotlights.
Mario & Luigi: Partners in Time mentions a Fishin' Lakitu-based business.
Mario Hoops 3-on-3
In Mario Hoops 3-on-3, Fishin' Lakitu will hold the ball on its fishing pole.
Mario Sports Mix
Fishin' Lakitu appears in basketball and hockey in Mario Sports Mix. In the former, it holds the ball on its fishing pole for the tip-off (just like in Mario Hoops), where at the start of a face-off in the latter, it holds the puck.
Paper Mario series
Paper Mario: Sticker Star
Paper Mario: Color Splash
Mario & Sonic at the Sochi 2014 Olympic Winter Games
Although regular Lakitus appear throughout the Mario & Sonic series as referees, Mario & Sonic at the Sochi 2014 Olympic Winter Games is the only game where a Fishin' Lakitu appears. He uses the fishing pole for starting lights in some events.
Appearances in other media
Super Smash Bros. series
Super Smash Bros.
Super Smash Bros. Melee
In Super Smash Bros. Melee, the Super Mario World incarnation of Fishin' Lakitu appears in the Super Mario World-themed Yoshi's Island stage. Also, Lakitu's trophy in this game depicts it with a fishing rod.
Super Smash Bros. Brawl
Super Smash Bros. for Nintendo 3DS / Wii U
Super Smash Bros. Melee's Yoshi's Island stage returns yet again in Super Smash Bros. for Wii U. The Peach's Castle stage from Super Smash Bros. also returns to both games later on as downloadable content.
Super Smash Bros. Ultimate
Fishin' Lakitu returns as background of Yoshi's Island stage from Super Smash Bros. Melee returns in Super Smash Bros. Ultimate.
Nintendo Badge Arcade
A hand-drawn version of Mario Kart 8's incarnation of Fishin' Lakitu appears in Nintendo Badge Arcade.
Names in other languages