Trainer Classes that focus on typeless categories

Bird Keeper -Focuses on bird-based Pokémon.
Scientist: Focuses on artificial Pokémon and humanoid Psychics.
Hex Maniac: Magical Pokémon, ie. Ghost and Psychic.
Guitarist: Sound-based Pokémon.
Fisherman: Focuses on fish-based Pokémon.
Scuba Diver, Free Diver: Uses deep sea Pokémon.
Beauty: Cute and beautiful Pokémon.
Poké Fan: Cute Pokémon, primarily Fairy Egg Group and Baby Pokémon.
Sightseer :Typical variant of a regional variant.

Ace Trainers and their variants, and regular human trainers(Fans, sportspeople etc.) are not listed.

Honorable mentions:

Super Nerd: Generally uses a Fire, Poison and Electric Pokémon, but also prefers synthetic Pokémon.

Trainer Classes according to types:

Bug-type specialists:
Bug Catcher: Ever-present in Japanese based regions, but doesn't appear in Western regions.
Bug Maniac: Hoenn exclusive.
Bug Catching Man: PBR exclusive

Dark-type specialists:
Delinquent: OR/AS exclusive
Roughneck: Doesn't appear in Hoenn and Kalos
Street Thug: OR/AS exclusive

Dragon-type specialists:
Dragon Tamer: Exclusive to Sinnoh and Hoenn.

Electric-type specialists:
Cameraman: Sinnoh exclusive.
Cyclist: Exclusive to Sinnoh and Unova.
Electrifying Guy: PBR exclusive
Engineer: Kanto exclusive
Guitarist: Doesn't appear in Kanto and Kalos
Poké-Kid: Sinnoh exclusive
Rocker: Kanto exclusive
Shocking Girl: PBR exclusive
Super Nerd: Kanto and Johto exclusive
Triathlete(Biker): Hoenn exclusive

Fairy-type specialists:
Fairy Tale Girl: Kalos and Hoenn exclusive
Furisode Girl: Kalos exclusive

Fighting-type specialists:
Battle Girl: Doesn't appear in Kanto and Johto
Black Belt: Everpresent.
Crush Girl: FR/LG exclusive
Crush Kin: FR/LG exclusive
Expert: Hoenn exclusive
Hiker: Everpresent
Linebacker: Unova exclusive.
Old Couple: Hoenn exclusive
Roughneck: Doesn't appear in Hoenn and Kalos
Sailor: Japanese regions only.

Fire-type specialists:
Burglar: Kanto and Johto exclusive
Cyclist: Exclusive to Sinnoh and Unova.
Firebreather: Johto exclusive
Kindler: Hoenn exclusive
Super Nerd: Kanto and Johto exclusive

Flying-type specialists:
Cyclist: Exclusive to Sinnoh and Unova.
Bird Keeper: Japanese regions only
Golfer: Alolan exclusive
Pilot: Unova only.
Sky Trainer: Kalos only.

Ghost-type specialists:
Channeler: Kanto exclusive
Hex Maniac: Hoenn and Kalos exclusive.
Medium: Johto exclusive

Grass-type specialists:
Aroma Lady: Hoenn and Sinnoh exclusive.
Pokémon Ranger: Doesn't appear in Kanto(besides Sevii Islands) and Johto.

Ground-type specialists:
Hiker: Everpresent
Muddy Boy: PBR exclusive
Pokémon Ranger: Doesn't appear in Kanto(besides Sevii Islands), Kalos, Unova and Johto.

Ice-type specialists:
Boarder: Johto exclusive
Cool Beauty: PBR exclusive
Icy Guy: PBR exclusive
Skier: Exclusive to Johto, and Sinnoh.

Normal-type specialists:
Painter: FR/LG exclusive
Tuber: Doesn't appear in Unova, Kalos and Kanto besides Sevii Islands.

Poison-type specialists:
Biker: Doesn't appear in Kalos, Hoenn and Sinnoh
Janitor: Unova exclusive
Ninja Boy: Hoenn and Sinnoh exclusive.
Super Nerd: Kanto and Johto exclusive

Psychic-type specialists:
Clown: Sinnoh exclusive
Hex Maniac: Hoenn and Kalos exclusive.
Psychic: Everpresent

Rock-type specialists:
Hiker: Everpresent

Steel-type specialists:
Steel Spirit: PBR exclusive

Water-type specialists:
Fisherman: Everpresent
Free Diver: OR/AS exclusive
Navigator: XD exclusive.
Sailor: Japanese regions only.
Scube Diver: OR/AS exclusive
Sis and Bro: Hoenn and Sevii Islands exclusive
Surfer: Alola exclusive.
Swimmer: Everpresent
Swimming Club Member: PBR exclusive
Triathlete (swimmer): Hoenn exclusive
Tuber: Doesn't appear in Unova, Kalos and Kanto besides Sevii Islands.

Edit(29.03.2018): Added USUM additions.