- Safari Ball in EP035 and "Showdown at the Po-Ké Corral".
The first time that a Poké Ball aside from the normal variation was seen was in EP035, where Ash was given 30 Safari Balls in order to compete in the Safari Game. With these 30 Safari Balls, Ash attempted to catch various rare Pokémon; however, he only managed to capture an entire herd of Tauros. They appeared in Safari Balls in Showdown at the Po-Ké Corral; however, whenever Ash uses one of his Tauros in a battle, it is sent out from a standard Poké Ball.
- GS Ball first debuted in Pallet Party Panic until Johto journeys and meeting Kurt.
The GS Ball was the second of the variant Poké Balls to appear in the anime, this time with a special purpose. This mysterious ball was unable to be opened by Professor Ivy, and served as the reason for Ash's journeys to the Orange Archipelago (to pick it up) and Johto (to deliver it to Kurt), so that what was contained within it could be discovered.
- Fast Ball in Going Apricorn! Lure Balls in Gettin' The Bugs Out, and used by Ash and Misty to catch Totodile and Corsola respectively. Heavy Ball in same episode, another one in Gulpin It Down. Kimono Sisters used to keep their Eeveevolutions in Apricorn balls.
Also related to Kurt, as in the games, the first non-standard Poké Ball variants, the Apricorn balls, made an appearance in the anime, and several were given to the members of the main cast. All three members of the main cast received a Fast Ball each in Going Apricorn!, with Brock using his to catch a Pineco shortly after receiving it. In the next episode, Brock received a Heavy Ball, while Ash and Misty received a Lure Ball each. Both Ash and Misty would use their Lure Balls to capture a Totodile and Corsola, respectively. Another Heavy Ball appeared in Gulpin It Down, where it was used to capture a giant Gulpin, though this was not the one belonging to Brock. In Trouble's Brewing, the Kimono Sisters (excluding Sakura) were shown keeping their Eeveelutions inside Apricorn Poké Balls, with Satsuki's Jolteon's ball being a Moon Ball, Sumomo's Vaporeon's ball being a Lure Ball, and Tamao's Umbreon's ball being a Fast Ball. While Koume's Flareon's ball wasn't shown, her kimono pattern indicates it being a Love Ball.
- Master Ball in "Whiscash and Ash" episode.
The Master Ball itself has only appeared once as an actual Poké Ball, in Whiscash and Ash, where it was used by Sullivan in a last resort attempt to catch a wild Whiscash called "Nero". Despite the fact that a Master Ball cannot be escaped from, the Whiscash swallowed the Master Ball, thus preventing capture, and disappeared back into the water.
- Gen III and IV specialty balls mostly appeared in cameos:
The Generation III specialty balls have mostly been seen in cameos. The Repeat Ball and Luxury Ball appeared in the opening of Jirachi: Wish Maker. These balls contained Brendan's Shiftry and Aggron, respectively.
The debut of most of the specialty balls, both from Generation III and IV, came in the ending Which One ~ Is It?, which contained the first appearance of the Great Ball and Ultra Ball, as well as the first anime appearance of the Premier, Heal, Net, Dusk, Nest, Quick, Timer, and Dive Balls.
- The following Poke Balls appeared physically for the first time in "A Frenzied Factory Fiasco!"
The first proper appearance of the Great, Ultra, Net, Nest, Dusk, Dive, Repeat, Premier, and Heal Balls was in A Frenzied Factory Fiasco!, where these balls were seen being manufactured at the Poké Ball Factory, in addition to normal Poké Balls, Luxury Balls, and Safari Balls, although the last ones were not shown.
- James' Poke ball collection was first revealed in Episode "First Catch in Alola, Ketchum-Style!".
It was revealed in First Catch in Alola, Ketchum-Style! that James had been collecting Poké Balls. When Jessie's attempts to catch a Mimikyu with typical Poké Balls had failed, she grabbed the Luxury Ball James had been polishing and used it instead, capturing Mimikyu, much to James's dismay. The rest of James's collection appeared in Acting True to Form!, where it was revealed to also contain a Great, Ultra, Premier, Dusk, Heal, and Quick Ball, marking the first proper anime appearance of a Quick Ball. The collection reappeared in Splash, Dash, and Smash for the Crown!, where it now consisted of a Premier, Dive, Nest, Quick, Heal, and Dusk Ball.
- Gladion's Pokemon were kept in Ultra Ball, Premier Ball, Heal Bell and Dusk Ball respectively.
Gladion keeps all of his known Pokémon in special Poké Balls. His Lycanroc is kept in an Ultra Ball (as first seen in A Glaring Rivalry!), his Silvally is kept in a Premier Ball (as first seen in Rising from the Ruins!), his Umbreon is kept in a Heal Ball (as first seen in Rescuing the Unwilling!), and his Zoroark is kept in a Dusk Ball (as first seen in the Poké Problem extra scene of SM127).
- Beast Balls made their debut in A Mission of Ultra Agency! and appeared in many Ultra Guardian episodes.
Beast Balls debuted in A Mission of Ultra Urgency!, in which the Aether Foundation provided multiple of them for Ash and his classmates to use during their missions as Ultra Guardians. Since then, Beast Balls have appeared in a number of episodes.
Another Ultra Ball appeared in The Legends Awaken! where it was used by Leon in a try to catch Eternatus.
In Sword and Shield... The Legends Awaken!, Leon used an Ultra Ball in an attempt to catch Eternatus, only for Eternatus to break out.
These are all I could find. Please comment if anything is missing or incorrect.
Hope this helps.