I'm answering this because I have nothing better to do, and I want Internet points! :D
Pokemon only seen in flashbacks, or Pokemon that were released/traded, still count as being had by a Trainer, for the purpose of this answer. A person that is caring for/protecting a Pokemon does not count as owning that Pokemon (for example, Shaymin was cared for, but never actually captured, so for this list, Shaymin is on here.).
Gen 1
Omanyte doesn't seem to have a specific Trainer, according to Bulbapedia.
Kabuto doesn't seem to have a specific Trainer either.
Mew
Gen 2
Unown
Porygon2
Raikou
Debatably Entei (Illusions of one were created by the character Molly Hale, but it was never technically captured)
Lugia befriended Ash and Goh, but was never captured
Gen 3
Lileep was looked after, but never used by a real Trainer.
Anorith was never used in battle.
Jirachi
Gen 4
Porygon-Z
Regigigas
Cresselia
Manaphy hatched with May, but hatching isn't technically capturing.
Arceus
Gen 5
Victini
Tympole
Cofagrigus
Archen was revived and raised by Fennel, but never captured or used.
The same with Archeops.
Cobalion
Terrakion
Kyurem was controlled by Team Plasma, but it may not have been captured.
Keldeo
Meloetta
Genesect
Gen 6
Skrelp
Amaura was revived, never captured.
Carbink
Trevenant
Zygarde was captured, but escaped.
Diancie
Hoopa
Volcanion
Gen 7
Cutiefly
Wishiwashis are at the Aether Paradise, they may have not been technically captured.
Toxapex
Dewpider
Bewear
Oranguru
Wimpod
Minior
Drampa
Dhelmise
Tapu Koko
Tapu Lele
Tapu Bulu
Tapu Fini
Cosmog
Cosmoem
Solgaleo
Lunala
Celesteela
Kartana
Guzzlord
Necrozma
Marshadow
Gen 8
Thwackey
Corvisquire
Blipbug
Orbeetle
Nickit
Theivul
Boltund
Rolycoly
Applin
Silicobra
Barraskewda
Toxtricity
Sizzlipede
Perrserker
Dracozolt
Arctovish
Drakloak
Zacian
Zamazenta
Kubfu
Zarude
Glastrier
Spectrier
Calyrex
Wyrdeer
Kleavor
Ursaluna
Basculegion
Sneasler
Overqwil
Enamorus
Gen 9
Sprigatito
Floragato
Meowscarada
Fuecoco
Crocalor
Skeledirge
Quaxly
Quaxwell
Quaquaval
Lechonk
Oinkologne
Tarountula
Spidops
Nymble
Lokix
Pawmi
Pawmo
Pawmot
Tandemaus
Maushold
Fidough
Dachsbun
Smoliv
Dolliv
Arboliva
Squawkabilly
Nacli
Naclstack
Garganacl
Charcadet
Armarouge
Ceruledge
Tadbulb
Bellibolt
Wattrel
Kilowattrel
Maschiff
Mabosstiff
Shroodle
Grafaiai
Bramblin
Brambleghast
Toedscool
Toedscruel
Klawf
Capsakid
Scovillain
Rellor
Rabsca
Flittle
Espathra
Tinkatink
Tinkatuff
Tinkaton
Wiglett
Wugtrio
Bombirdier
Finizen
Palafin
Varoom
Revavroom
Cyclizar
Orthworm
Glimmet
Glimmora
Greavard
Houndstone
Flamigo
Cetoddle
Cetitan
Veluza
Dondozo
Tatsugiri
Annihilape
Clodsire
Farigiraf
Dudunsparce
Kingambit
Great Tusk
Scream Tail
Brute Bonnet
Flutter Mane
Slither Wing
Sandy Shocks
Iron Treads
Iron Bundle
Iron Hands
Iron Jugulis
Iron Moth
Iron Thorns
Frigibax
Arctibax
Baxcalibur
Gimmighoul
Gholdengo
Wo-Chien
Chien-Pao
Ting-Lu
Chi-Yu
Roaring Moon
Iron Valiant
Koraidon
Miraidon
Hope I helped!
Source: Every single Pokemon's Bulbapedia article