- Meloetta (Normal/Fighting in Pirouette Form, signature in gen V only): Relic Song (move)
- Bibarel (Normal/Water): Unaware (ability, signature in gen IV only)
- Sawsbuck (Normal/Grass): Horn Leech (move, signature in gen V only)
- Drampa (Normal/Dragon): Berserk (ability, signature in gen VII only)
- Hawlucha (Fighting/Flying, signature prior to gen VIII): Flying Press (move)
- Marshadow (Fighting/Ghost): Spectral Thief (move)
- Meditite family (Fighting/Psychic, signature in gen III only): Pure Power (ability)
- Crabominable (Fighting/Ice): Ice Hammer (move)
- Salandit family (Poison/Fire): Corrosion (ability)
- Slowbro (Poison/Psychic in Galarian Form): Quick Draw (ability), Shell Side Arm (move)
- Slowking (Poison/Psychic in Galarian Form): Curious Medicine (ability), Eerie Spell (move)
- Weezing (Poison/Fairy in Galarian Form): Strange Steam (move)
- Geodude family (Rock/Electric in Alolan Form): Galvanize (ability)
- Tyranitar (Rock/Dark): Sand Stream (ability, signature in gen III only)
- Heracross (Bug/Fighting, signature prior to gen VII): Megahorn (move, signature in gen II only)
- Shedinja (Bug/Ghost): Wonder Guard (ability)
- Volcarona (Bug/Fire, signature prior to gen VIII): Fiery Dance (move)
- Paras family (Bug/Grass, signature prior to gen IV): Spore (move, signature prior to gen III)
- Ribombee (Bug/Fairy): Polen Puff (move, signature in gen VII only)
- Drifloon family (Ghost/Flying, signature prior to gen VII): Flare Boost (ability)
- Gastly family (Ghost/Poison): Night Shade (move, signature in gen I only)
- Giratina (Ghost/Dragon, signature prior to gen VIII): Shadow Force (move)
- Mimikyu (Ghost/Fairy): Disguise (ability)
- Aegislash (Steel/Ghost): Stance Change (ability), King's Shield (move)
- Dialga (Steel/Dragon, signature prior to gen VIII): Roar of Time (move)
- Litleo family (Fire/Normal): Noble Roar (move, signature in gen VI only)
- Blaziken (Fire/Fighting, signature in gen III only): Blaze Kick (move, signature in gen III only)
- Heatran (Fire/Steel): Magma Storm (move)
- Volcanion (Fire/Water): Steam Eruption (move)
- Lotad family (Water/Grass): Rain Dish (ability, signature prior to gen V)
- (Note that while Marill and Azumarill have had a unique type combination and a signature ability, neither quality was unique at the same time due to the Fairy type not existing prior to gen VI)
- Ferroseed family (Grass/Steel, signature prior to gen VII): Iron Barbs (ability, signature prior to gen VII)
- Exeggcute family (Grass/Psychic, signature in gen I only): Barrage (move)
- Snover family (Grass/Ice): Snow Warning (ability, signature prior to gen VI)
- Helioptile family (Electric/Normal): Electrify (move, signature prior to gen VIII)
- Toxtricity (Electric/Poison): Punk Rock (ability), Overdrive (move)
- Raichu (Electric/Psychic in Alolan Form): Surge Surfer (ability)
- Morpeko (Electric/Dark): Hunger Switch (ability), Aura Wheel (move)
- Frosmoth (Ice/Bug): Ice Scales (ability)
- Jynx (Ice/Psychic, signature prior to gen VIII): Lovely Kiss (move)
- Kommo-o (Dragon/Fighting): Clanging Scales (move), Clangorous Soul (move)
- Reshiram (Dragon/Fire, signature in gen V only): Blue Flare (move)
- Zekrom (Dragon/Electric, signature in gen V only): Bolt Strike (move)
- Kyurem (Dragon/Ice): Glaciate (move), Freeze Shock (move, signature for Black Kyurem), Ice Burn (move, signature for White Kyurem)
- (Note that while Sableye has had a unique type combination and a signature ability, neither quality was unique at the same time due to the introduction of Spiritomb in the same generation Sableye was given the signature ability)
- Impidimp family (Dark/Fairy): Confide (move), False Surrender (move, signature for Morgrem and Grimsnarl), Spirit Break (move, signature for Grimsnarl)
Note that this list does not include moves or abilities on prior evolutions who lack the type. (Volt Tackle is not listed on Alolan Raichu because the prior evolutions who are pure Electric learn it too) This list also does not consider signature moves bulbapedia classifies as due to only one Pokemon learning by level up where the move is also an egg move. It does include signature moves for entire evolution lines with the type combo that are only on the final form.
Source: checking [this][1] with [this][2] and [this][3], making sure to remove examples where signature moves can be learned by breeding and signature abilities on evolution families where the type changes.
Hope this helped!
[1]: https://bulbapedia.bulbagarden.net/wiki/List_of_Pokémon_with_unique_type_combinations
[2]: https://bulbapedia.bulbagarden.net/wiki/Signature_move
[3]: https://bulbapedia.bulbagarden.net/wiki/Signature_Ability