I made a list of all legal egg moves as of pre-bank for the Alola pokemon + Alola forms. For pokemon in the field egg group, all egg moves are possible as Smeargle is catchable in the Alola dex.
Rowlet (Flying): Haze/Confuse Ray: Zubat; Defog: Rufflet; Baton Pass: Oricorio -Illegal: Curse (Farfetch'd); Ominous Wind (Xatu)
Litten (Field): Nasty Plot/Body Slam/Crunch/Fake Out/Revenge/Heat Wave
Popplio (Water 1/Field): Charm/Amnesia/Aqua Ring/Aromatic Mist/Perish Song/Wonder Room
Pikipek (Flying): Tailwind/Brave Bird: Rufflet; Mirror Move: Spearow -Illegal: Boomburst (Taillow/Noibat/Chatot), Uproar(Hoothoot/Spearow/Doduo)
Yungoos (Field): Revenge/Last Resort
Grubbin (Bug): Harden: Pinsir; Mud Shot: Surskit (bred w/Poliwag for Mud Shot) - Illegal: Electroweb (Galvantula); Endure (Heracross)
Crabrawler (Water 3): Wide Guard: Tirtouga; Endeavour: Archen -Illegal: Amnesia (Krabby/Lileep); Superpower (Requires ORAS move tutor Water 3 compatible pokemon)
Oricorio (Flying): Pluck: Spearow, Tailwind: Rufflet, Safeguard: It's a TM -Illegal: Captivate (Woobat)
Cutiefly (Bug/Fairy): Baton Pass: Ledyba; Speed Swap: A-Raichu; Moonblast/Bestow: Clefairy -Illegal: Skill Swap (Spritzee)
Rockruff (Field): Fire Fang/Thunder Fang/Crush Claw/Sucker Punch/Thrash
Wishiwashi (Water 2): Mist/Water Sport/Water Pulse: Wailmer; Muddy Water: Barboach
Mareanie (Water 1): Stockpile/Swallow/Spit Up: Shellos (bred w/Drifloon for moves); Haze: Surskit
Mudbray (Field): Body Slam/Double-Edge/Magnitude/Close Combat/Mud Bomb
Dewpider (Water 1/Bug): Stockpile/Spit Up: Shellos -Illegal: Power Split (Shuckle) -Legal via QR: Aurora Beam (Spheal)
Formantis (Grass): Aromatherapy/Giga Drain/Leaf Storm: Bellossom -Illegal: Defog (Requires TM with Grass compatible from D/P/PT) -Legal via QR: Weather Ball (Bellsprout/Roselia)
Morelull (Grass): Posion Powder/Stun Spore/Leech Seed: Exeggcute -Illegal: Amnesia (Bulbasaur + Slowpoke required)
Salandit (Monster/Dragon): -Illegal: Belch (Ekans); Fake Out (Scraggy/Squirtle + Smeargle required); Knock Off (Requires ORAS move tutor Monster/Dragon compatible pokemon)
Stufful (Field): Ice Punch/Thunder Punch/Force Palm/Endure/Mega Kick/Stomping Tantrum
Bounsweet (Grass): Synthesis/Play Rough: Comfey; Charm: Cottonee -Illegal: Acupressure (Maractus), Feint (Breloom)
Comfey (Grass): Lucky Chant: Oddish; After You: Petilil -Illegal: Amnesia (Bulbasaur + Slowpoke required), Endure (Bulbasaur + Shieldon/Paras + Heracross required)
Oranguru (Field): Extrasensory/Wonder Room/Psychic Terrain
Passimian (Field): Siesmic Toss/Vital Throw/Quick Guard/Iron Head/Quick Attack/Feint
Wimpod (Bug/Water 3): Spikes: Shellder; Aqua Jet/Wide Guard: Tirtouga, Harden: Pinsir -Illegal: Metal Claw (Krabby)
Sandyghast (Amorphous): Amnesia/Stockpile/Swallow/Spit Up: Drifloon; Destiny Bond: Ghastly -Illegal: Ancient Power (Lileep)
Pyukumuku (Water 1): Endure: Poliwag (bred w/ Corsola for Endure); Venom Drench: Mareanie, -Illegal: Bestow (Passed from Zizagoon -> Delibird -> Froakie); Tickle (Omanyte)
Komala (Field): Charm/Wish/Play Rough/Sing
Turtonator (Monster/Dragon): Wide Guard: Tirtouga; Head Smash: Cranidos, -Illegal: Fire Spin (Charmander); Revenge (Druddigon)
Togedemaru (Field/Fairy): Reversal/Present/Encore/Twineedle/Wish/Fake Out/Tickle/Flail/Disarming Voice
Mimikyu (Amorphous): Grudge/Nightmare/Destiny Bond/Curse: Ghastly (+ Misdreavus for Grudge)
Bruxish (Water 2): Water Pulse: Wailmer; Poison Fang/Ice Fang/Rage: Carvanha
Drampa (Monster/Dragon): Hurricane/Mist/Dragon Rush: Dragnonite (bred w/Wingull/Surskit for Mist) -Illegal: Play Rough (Skrelp), Razor Wind (Tropius + Absol required)
Jangmo-o (Dragon): Dragon Breath: Gible -Illegal: Counter (Goomy + Scraggy required) -Legal via QR: Reversal (Axew + Ryhorn required)
A-Rattata (Field): Snatch/Stockpile/Fury Swipes/Switcheroo/Counter/Reversal/Uproar/Swallow/Me First/Revenge/Final Gambit
A-Pikachu (Field): Bide/Reversal/Present/Encore/Wish/Charge/Double Slap/Fake Out/Thunder Punch/Tickle/Flail/Endure/Lucky Chant/Bestow/Disarming Voice/Electric Terrain
A-Sandshrew (Field): Flail/Counter/Icicle Spear/Metal Claw/Crush Claw/Night Slash/Icicle Crash/Endure/Chip Away/Curse/Amnesia
A-Vulpix (Field): Freeze-Dry/Hypnosis/Flail/Spite/DisableHowl/Agility/Encore/Moonblast/Power Swap/Secret Power/Charm/Tail Slap/Extrasensory
A-Meowth (Field): Hynosis/Assist/Foul Play/Spite/Charm/Amnesia/Parting Shot/Flail/Flatter/Punishment/Snatch/Covet
A-Geodude (Mineral): Block/Wide Guard/Magnet Rise: Probopass; Flail/Counter: Sudowoodo -Illegal: Endure (Dwebble + Heracross/Yamask + Stunfisk required); Rock Climb (Ferrothorn); Curse/Screech (Steelix) -Legal via QR: Autotomize (Honedge)
A-Grimer (Amorphous): Spite/Curse/Mean Look: Ghastly; Stockpile/Swallow/Spit Up: Drifloon; Imprison: Misdreavus -Illegal: Shadow Sneak/Scary Face (Pumpkaboo); Clear Smog/Assurance (Weezing); Pursuit (Duskull)
A-Exeggutor (Grass): Lucky Chant/Moonlight/Giga Drain/Natural Gift/Grassy Terrain: Gloom; Leaf Storm: Bellossom -Illegal: Block (Tangela); Power Swap/Nature Power/Synthesis (Seedot bred w/Smeargle for Power Swap); Curse (Ferroseed); Skill Swap (Requires Exeggutor from previous gen w/Skill Swap) -Legal via QR: Ingrain (Roselia), Ancient Power (Chikorita bred w/Shieldon for move)
A-Marowak (Monster): Ancient Power/Iron Head/Endure: Shieldon; Perish Song: Lapras; Chip Away/Screech: Cranidos -Illegal: Detect (Treecko); Double Kick (Nidoran-M); Skull Bash (Squirtle)
A-Diglett (Field): Feint Attack/Metal Sound/Ancient Power/Pursuit/Beat Up/Thrash/Reversal/Headbutt/Endure/Final Gambit/Memento