How to teach a Pokémon new abilities?

Question

As the title already states: What are all the ways in which a Pokémon can gain new abilities? Aside from level-up. Thus, methods that would mean it ends up with more and/or other abilities than it'd naturally have.

Are you interested in all possible methods of doing this, including general methods available to all trainers and also ones that are restricted to specific trainer classes? Also which version of PTU are you using (probably 1.05 but perhaps the 2.0 playtest material)? Are you interested in things from the expansions as well such as Game of Throhs, The Blessed and the Damned, or Do Porygon Dream of Mareep? — Exempt-Medic, Jul 14 '20 at 21:34
Right now, I am playing 1.05 indeed. I am interested in all methods. I'm not using any of the expansions, but if people are willing to add those into their answer anyway: Feel free, but mark that section accordingly. I'm sure somebody who is not me might be very glad to find that info here. — Arthaban, Jul 14 '20 at 21:39

score 4 · Accepted Answer · edited Aug 30 '20 at 21:31

General Methods

Ability Mastery

This Poké Edge (available after a Pokémon reaches level 60) lets you spend 3 Tutor Points to grant your Pokémon an additional Ability.

Full Incense

This grants the holder the Stall Ability.

Trainer Class Specific

Mentor

Mentor Lessons: Versatile Teachings lets you spend 1 Tutor Point to swap your Pokémon's Basic Ability for a different Basic Ability or an Advanced Ability for a different Basic or Advanced Ability.

Cheerleader

Cheer Brigade lets you spend 2 Tutor Points to give your Pokémon the Friend Guard Ability.

Duelist

Effective Methods lets you spend 2 Tutor Points to give your Pokémon either the Exploit or Tolerance Ability.

Duelist's Manual lets you switch which of those two Abilities a Pokémon has until the end of their next turn.

Rider

Ramming Speed lets you spend 2 Tutor Points to give your Pokémon the Run Up Ability.

Taskmaster

Savage Strike lets you spend 2 Tutor Points to give your Pokémon the Cruelty Ability.

Stat Ace

Stat Embodiment grants your Pokémon an Ability until the end of the Scene. Which Ability you grant depends on which Stat you chose when taking this Trainer Class.

Style Expert

Style Entrainment grants your Pokémon an Ability until the end of the Scene. Which Ability you grant depends on which Contest Stat you chose when taking this Trainer Class.

Type Ace

Type Ace lets you spend 2 Tutor Points to grant your Pokémon either the Type Strategist or Last Chance Ability. The latter's effect depends on which Type you chose when taking this Trainer Class.

Normal Ace

Extra Ordinary lets you grant a Pokémon either the Last Chance of Normal Strategist Ability (whichever it does not currently have).

Chef

Culinary Appreciation lets you spend 2 Tutor Points to grant your Pokémon the Gluttony Ability.

Chronicler

Targeted Profiling lets you grant your Pokémon the Mold Breaker Ability against all Pokémon and Trainers in your Archive.

Observation Party lets you grant your Pokémon either the Keen Eye or Perception Ability when in a Location in your Archive. Which Ability is given is chosen when first getting your Travel Archive.

General Researcher

Instant Analysis lets your spend 2 Tutor Points to give your Pokémon the Forewarn Ability.

Paleontology Researcher

Fossil Restoration lets you cause a reviving Fossil to have 2 fewer Tutor Points and start with an extra Basic Ability (or Advanced Ability if one does not exist). This explicitly does not affect Abilities gained later on.

Hunter

Pack Tactics lets you spend 2 Tutor Points to grant your Pokémon either the Pack Hunt or Teamwork Ability.

Pokémon Specific

Mega Evolution

Several Pokémon can Mega Evolve so I won't list them all, but doing so grants them a new Ability.

Trace

This Ability (available to Espeon, the Porygon line, the Ralts line, and Mega Alakazam) grants you an Ability that a nearby Trainer or Pokémon has.

Thick Club

Grants specifically Cubone or Marowak holding this the Pure Power Ability.

The Sewaddle Line (Sewaddle, Swadloon, Leavanny)

These Pokémon all get the Leaf Gift Ability which grants a set of two Abilities depending on which Leaf Suit they made that day.

The Yamask Line (Yamask, Cofagrigus)

These Pokémon all get the Mummy Ability. When hit by a Melee Attack the Attacker's Abilities are all replaced with Mummy for the encounter.

Furfrou

Furfrou's Ability depends on what hairstyle they have which can be changed at a hair parlor.

Rotom

The Poltergeist Ability causes each Rotom form to have its own Ability.

Move Specific

Magnet Rise

Grants the Levitate Ability for 5 turns.

Worry Seed

Changes one of your target's Abilities into Insomnia for the encounter.

Entrainment

Grants one of your Abilities to your target for 3 turns.

Simple Beam

Changes one of your target's Abilities into Simple.

Transform

Grants you the target's Abilities.

Role Play

You gain one of your Target's Abilities for the encounter.

Skill Swap

You and your target swap an Ability for the encounter.

Telekinesis

Grants your Target the Levitate Ability while they are Lifted.

Other

The Abilities available to an Arbok are determined by its Serpent's Mask Ability.

The example Gym Battle where Badges are held items has the Summit Badge grant the Snow Cloak Ability and the Overcloak item which grants the Overcoat Ability.