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.).
Gens 1-5
Gen 1
Porygon was never technically under a Trainer's command, just a scientist and even then, only in a banned episode.
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
Sunkern (Ash evolved one, but never actually caught it)
Unown
Porygon2
Raikou
Debatably Entei (Illusions of one were created by the character Molly Hale, but it was never technically captured)
Suicune
Lugia befriended Ash and Goh, but was never captured
Gen 3
Lileep was looked after, but never used by a real Trainer.
Cradily was looked after as well, but never used by a Trainer.
Anorith was never used in battle.
Jirachi
Gen 4
Spiritomb
Porygon-Z
Regigigas
Cresselia
Manaphy
Arceus
Gen 5
Victini
Tympole
Cofagrigus
Tirtouga was befriended by someone, but not technically captured.
The same goes for Carracosta.
Archen was revived and raised by Fennel, but never captured or used.
The same with Archeops.
Ferroseed
Durant
Volcarona
Cobalion
Terrakion
Virizion
Kyurem
Keldeo
Meloetta
Genesect
Gen 6
Quilladin
Skrelp
Dragalge
Tyrantrum is owned by Scientists, but never stated to be captured.
Amaura was revived, never captured.
Aurorus was revived, never captured.
Carbink
Trevenant
Xerneas
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
Palossand is at the Aether Paradise, may not have been captured.
Minior
Bruxish
Drampa
Dhelmise
Jangmo-o
Tapu Koko
Tapu Lele
Tapu Bulu
Tapu Fini
Cosmog
Cosmoem
Solgaleo
Lunala
Nihilego
Celesteela
Kartana
Guzzlord
Necrozma
Marshadow
Gen 8
Grookey
Thwackey
Rillaboom
Cinderace
Drizzile
Inteleon
Corvisquire
Blipbug
Dottler
Orbeetle
Nickit
Theivul
Gossifleur
Eldegoss
Dubwool
Drednaw
Boltund
Rolycoly
Applin
Flapple
Appletun
Silicobra
Sandaconda
Cramorant
Arrokuda
Barraskewda
Toxel
Toxtricity
Sizzlipede
Centiskorch
Clobbopus
Grapploct
Sinistea
Hattena
Hattrem
Hatterene
Impidimp
Morgrem
Grimmsnarl
Obstagoon
Perrserker
Cursola
Sirfetch'd
Runerigus
Falinks
Pincurchin
Snom
Frosmoth
Stonjourner
Eiscue
Cufant
Dracozolt
Arctozolt
Dracovish
Arctovish
Dreepy
Drakloak
Dragapult
Zacian
Zamazenta
Eternatus
Kubfu
Urshifu
Zarude
Calyrex
Regieleki
Regidrago
Wow, that took a really long time. Hope I helped!
My source is every single Pokemon's Bulbapedia article.