Articuno
15 Pokémon including the following specialties: Build, Crush, Fly, Water.
All 14 Legendary and Mythical Pokémon in Pokémon Pokopia, and exactly what each one takes to bring home. None of them arrive by building a habitat.
None of these are attracted by a habitat. Each has its own quest.
Every ordinary Pokémon in Pokopia arrives because you built the thing it likes. Legendaries ignore that entirely. Not one of the fourteen is attracted by a habitat, so no amount of building will bring them in, and each has its own way of turning up instead.
There is a practical reason to chase them beyond collecting. A legendary pulls the Environment level of wherever it settles up much faster than a normal resident, so raising one legendary's comfort lifts the whole area with it.
The birds are the most straightforward of the fourteen, because they are just builds. Three special kits are sitting around Palette Town waiting to be found, and each wants 15 Pokémon on the job plus a serious pile of materials. Put one up and its bird is inside.
The Freezing Chambers gives you Articuno, the Abandoned Power Plant gives you Zapdos, and the Altar of the Flame gives you Moltres. Full material costs for all three are in the table below.
Raikou, Entei, Suicune, Mewtwo and Phione are all met on Dream Islands rather than at home. Setting out with a particular doll leans the trip toward a particular one, but it only leans it. You will be going back more than once.
Once you have talked to it on its island you can bring it home and give it somewhere to live, exactly like any other resident.
You will see both of these pass overhead long before you can do anything about it. Leave them alone and they drop a feather on the way out: Ho-Oh a Rainbow Feather, Lugia a Silver one.
The bells that call them down come from the other legendaries, and they cross over. Befriending all three birds earns you the Tidal Bell recipe, which takes 5 Rare Pokémetal and 5 Silver Feather. Befriending all three beasts earns the Clear Bell, at 5 Rare Pokémetal and 5 Rainbow Feather. So the birds get you the bell for Lugia's feather, and the beasts get you the one for Ho-Oh's.
With a bell made, wait for the right silhouette to pass, put the bell down and ring it.
Volcanion is befriended as part of finishing Rocky Ridges, but befriending it is not the same as it moving in. For that you have to hold a second party in the same place and get the mood back up to 100.
The timing is the part people get wrong, and it helps to keep the two parties apart. Volcanion does turn up during the first one, appearing at the top of the volcano to give the celebration its blessing, but it says plainly that it is not interested in joining you. Joining happens after the second party: wait for that one to run its course, then load a cannon and set off fireworks, and Volcanion comes back asking to move in on the grounds that your parties look like a good time. It settles in Rocky Ridges.
Kyogre is not found so much as summoned by accident. Restoring the Withered Wastelands runs into the problem that it has not rained there in living memory, and the fix is Slowpoke, whose yawn brings rain once the air is damp enough.
Getting the humidity to 100 means working the area over: Water Gun on every bit of standing water, rocks broken open to let rivers through, flower patches planted wherever they will go. Then Slowpoke yawns and you get a brief, disappointing drizzle.
What matters is the noise from the coast straight afterwards. Kyogre has heard the whole attempt and come to see, and it rains properly, soaking the area in one go.
Manaphy turns up twice. You help it at Bleak Beach first, which is what opens up diving for everyone in the Version 2.0 update, and it only moves in later once Popplio's Ocean Temple is standing.
The temple is a big build. It takes fifteen Pokémon in all, and six of those Popplio describes rather than names: Corsola, Mareanie, Corphish, Totodile, Barboach and Frillish. Frillish is the awkward one, since it will not help until you take it through the Cerulean Cave with a Dowsing Machine and dig its Starmie Doll out of the rocks by a Crystal Cluster.
Finish the temple, walk inside, and Manaphy is there. It stays in the area afterwards, and says the visit is thanks for the help back at the beach.
Mew is the long one, and it is a collection job rather than a quest. Twenty-seven Mysterious Slates are buried across the island in shiny patches of ground, and they surface at random, so there is no route to walk and no way to hurry it.
When you finally have all 27, carry them to the ruins near the Pokémon Center in the Withered Wastelands. Each slate belongs on one tile, and laying them all correctly completes a picture of Mew. Finish it and Mew can be found.
The doll you start a journey with biases which legendary you can meet. It is not a guarantee.
Each needs 15 Pokémon helping and a serious pile of materials.