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, Archen was cared for, but never actually captured, so for this list, Archen is on here.). This answer also doesn't differentiate between regional forms.
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
Lugia befriended Ash and Goh, but was never captured
Ho-oh
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
Diancie
Hoopa
Volcanion
Gen 7
Wishiwashi 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
Lunala
Celesteela
Kartana
Guzzlord
Necrozma
Marshadow
Gen 8
Thwackey
Corvisquire
Blipbug
Nickit
Thievul
Boltund
Applin
Barraskewda
Sizzlipede
Perrserker
Dracozolt
Arctovish
Drakloak
Zacian
Zamazenta
Kubfu
Zarude
Glastrier
Spectrier
Calyrex
Wyrdeer
Ursaluna
Basculegion
Sneasler
Overqwil
Enamorus
Gen 9
Spidops
Nymble
Tandemaus
Maushold
Nacli
Naclstack
Armarouge
Maschiff
Mabosstiff
Shroodle
Grafaiai
Toedscool
Toedscruel
Klawf
Capsakid
Rellor
Rabsca
Tinkatuff
Bombirdier
Palafin
Varoom
Revavroom
Orthworm
Glimmora
Greavard
Veluza
Farigiraf
Great Tusk
Scream Tail
Brute Bonnet
Flutter Mane
Slither Wing
Sandy Shocks
Iron Treads
Iron Hands
Iron Jugulis
Iron Moth
Iron Thorns
Frigibax
Wo-Chien
Chien-Pao
Ting-Lu
Chi-Yu
Roaring Moon
Iron Valiant
Miraidon
Walking Wake
Iron Leaves
Dipplin
Poltchageist
Okidogi
Munkidori
Fezandipiti
Ogerpon
Archaludon
Hydrapple
Gouging Fire
Raging Bolt
Iron Boulder
Iron Crown
Pecharunt
Hope I helped!
Source: Every single Pokemon's Bulbapedia article