POKÉMON PARTNERS
Most people in Bill's world also capture and tame Pokémon, regardless of whether or not they actually consider themselves trainers. As a collector, Bill is really no different. These are some of the Pokémon he's been known to have:
Game
Eevee: It's a well-known fact that Bill gives away several Eevee throughout the franchise. First, an Eevee is accidentally sent to him while he's adjusting the Time Capsule (wi-fi system in HGSS), which he gives to the Johto player character because he can't take care of it himself. (He states it's because he doesn't like being outdoors, which would be a problem for an active Pokémon like Eevee.) Second, there's an Eevee that he gave to Bebe, who subsequently gives it to the player in DPPt. Bebe notes that he's bred countless Eevee in the past, but it's not known how many he still has.
Abra: In Crystal, FRLG, and HGSS, Bill's fans (Brent and Celio) tell the player that the first Pokémon he apparently captured was an Abra.
Flareon, Jolteon, Vaporeon: These three Pokémon appear on Bill's PC after you return him to his normal self in RBY and FRLG. He calls them part of his collection and lists them among his favorite Pokémon, implying that he actually owns at least one of each.
Clefairy: In FRLG, Bill uses a Clefairy to test his Pokémon Teleporter. It doesn't end well.
Note: The Pokémon he uses in Gen I is not identified. It should also be noted that the Gen I Pokémon doesn't share an overworld sprite with normal Clefairy. This either means Bill used a different Pokémon in Gen I or that the combination of him and Clefairy can't be identified as either species.
Anime
Strangely, despite being a researcher, Bill has no known Pokémon of his own. In fact, when asked if he would capture the giant Pokémon (i.e., Dragonite) that he wants to study, he actually refuses, stating that just meeting it would be enough for him.
Toshihiro Ono Series
Venusaur: In "Clefairy Tale," Bill describes the evolution of his Ivysaur into a Venusaur as his inspiration for becoming a researcher. It's not known whether or not he still owns it.Slowpoke: At the end of the series, in the side story, Bill pulls a Slowpoke out of an unused databank and puts it on display in the Cutting-Edge Technology Research Center's annual open house. He has no idea that it was actually created by Team Rocket and left there because its very existence was the result of a embarrassingly botched experiment. Technically, the Slowpoke was still wild; it's just counted here because it was temporarily in Bill's possession until Team Rocket tried to steal it back.
Pokémon Special
Rattata: Technically, the first of Bill's Pokémon to be seen. In his introduction chapter ("Vs. Fearow"/"But Fearow Itself!"), echoing the events of the game, Bill uses a Rattata to test his teleporter. It also doesn't end well.
Vulpix: Besides Rattata, Vulpix is the first Pokémon confirmed to be one of Bill's. Its main duty is to serve as a kind of watchdog, a guardian to ward off burglars with its Flamethrower. Its first appearance was in Chapter 59 ("Vs Hitmonlee"/"Hitmonlee, Baby One More Time"), and since then, it and Exeggcute/Exeggutor have served as Bill's primary fighting force.
Magikarp: Bill's Magikarp had only one appearance, Chapter 74 ("Vs Magikarp"/"Make Way For Magikarp"). In this chapter, Bill, on the tail of the Elite Four, falls off the back of a wild Onix and into a pond. (It makes sense in context.) However, a second later, he realizes he isn't standing in water at all and uses Magikarp as a deus ex machina to Splash himself to safety. It's lucky that he did; the liquid was actually acid. (No, really.)
Exeggcute/Exeggutor: The egg Pokémon's first appearance was in Chapter 75 ("Vs Electrode"/"Electrode's Big Shock"). Cornered by Bruno, Bill and Lt. Surge struggle to come up with a way to end the battle quickly. Surge's strategy? Having Bill's Exeggcute plant Barrage all over the Onix to be detonated by his Electrode. From then on, Exeggcute made minor appearances until it evolved in the FRLG arc, where Bill used it to test the VS Seeker.
Farfetch'd: The last of Bill's Pokémon to appear, this time in the Crystal arc (chapter 119, "Vs Flaaffy"). Unlike most of his other Pokémon, Farfetch'd isn't used for battling or experiments. Instead, Bill uses it almost exclusively as a traveling companion and mode of transportation (by cramming himself onto its back and having it use Fly). As such, its first appearance showcased this purpose when Bill tried to use it to get away from a group of overly playful Staryu.
How I Became a Pokémon Card
Dratini: As a child, Bill stopped a group of townspeople from killing a lost and injured Dratini. The townsfolk look to a wise professor (read: Professor Oak) for advice, who then gives Bill the Pokémon to nurse back to health. The two grow close in the short time they spend together until Dratini's father, a massive Dragonite, tries to level the town in search of his son. Bill guides the Dratini back to the Dragonite and promises to never forget the serpent. It's implied in an epilogue at the end of the volume that this fed his subsequent interest in Pokémon.
Golden Boys
Espeon and Umbreon: These Pokémon were lent to the main characters Gold and Black (respectively) in order to help fight Team Rocket. Not much else is known about these Pokémon.