Trade Evolution (some require items, some don't)
Level Up Evolution
Leveling Up While Knowing A Certain Move
Leveling Up In A Certain Area
Evolving In A Certain Region (like giving Pikachu a Thunder Stone in Alola)
At A Certain Time In The Day
Using An Item (Fire Stone, Razor Claw, etc.)
Evolving A Pokemon With A Certain Gender (for example Burmy or Combee)
Evoling In A Certain Game (like Rockruff)
Unique Evolution Methods:
Galarian Farfetch'd evolves into Sirfetch'd after it lands three critical hits in a single battle.
Galarian Yamask evolves into Runerigus when the player travels through the stone gate in the Dusty Bowl after it has lost 49 HP or more from a single attack and did not faint in the same battle or since.
Milcery evolves into Alcremie when the player spins and strikes a pose while Milcery holds a Sweet. Unless Milcery is Shiny, Alcremie's color depends on the time of day, the length of the spin, and the direction of the spin; if Milcery is Shiny, Alcremie takes on its sole Shiny coloration. Alcremie's Sweet corresponds to the Sweet Milcery held.
Vanilla Cream: Spin clockwise quickly in the day
Ruby Cream: Spin anti-clockwise quickly in the day
Matcha Cream: Spin clockwise quickly at night
Salted Cream: Spin anti-clockwise quickly at night
Lemon Cream: Spin clockwise for over 5 seconds at night
Ruby Swirl: Spin anti-clockwise for over 5 seconds in the day
Caramel Swirl: Spin clockwise for over 5 seconds in the day
Mint Cream: Spin anti-clockwise for over 5 seconds at night
Rainbow Swirl: Spin anti-clockwise for over 10 seconds at dusk
Kubfu evolves into Urshifu when shown the scroll in one of the Towers of Two Fists. The form it evolves into depends on the tower it is trained in.
Kubfu evolves into Urshifu Single Strike Style when trained in the Tower of Darkness.
Kubfu evolves into Urshifu Rapid Strike Style when trained in the Tower of Waters.
Mantyke evolves into Mantine when it levels up if there is a Remoraid in the player's party.
Feebas evolves into Milotic when it levels up if its Beauty condition is 170 or above. From Generation V onward, it can also evolve into Milotic when traded while holding a Prism Scale.
When Nincada evolves into Ninjask (which it does starting at level 20), if there is an empty slot in the party, a Shedinja will appear. From Generation IV onward, the player must also have a spare standard Poké Ball in their Bag.
Pancham evolves into Pangoro starting at level 32, if there is a Dark-type Pokémon in the party.
Inkay evolves into Malamar when it levels up starting at level 30, if the player's system is held upside-down when it levels up. On the Nintendo Switch, this can only be accomplished in handheld mode.
Sliggoo evolves into Goodra starting at level 50, if it is raining or (in Generation VII) foggy in the overworld. (Rain that is only present in battle will not trigger the evolution.)
Burmy evolves starting at level 20, but its evolved form is determined by its gender and, if female, its cloak. As Burmy, its cloak changes according to the place where it last battled but has no effect on Burmy itself; once it evolves into Wormadam, its cloak no longer changes, and its types and the moves it can learn are dependent on its cloak.
Male Burmy → Mothim
Female Burmy → Wormadam
Female Plant Cloak Burmy → Plant Cloak Wormadam (Bug/Grass)
Female Sandy Cloak Burmy → Sandy Cloak Wormadam (Bug/Ground)
Female Trash Cloak Burmy → Trash Cloak Wormadam (Bug/Steel)
Tyrogue evolves starting at level 20, depending on its stats.
Hitmonlee if Attack > Defense
Hitmonchan if Attack < Defense
Hitmontop if Attack = Defense
Wurmple evolves into Silcoon or Cascoon starting at level 7, depending on its personality value. It is a hidden value determined when the Pokémon is caught and cannot be affected by the player.
Hope this helps! :)