Abilities According to Distribution:

All:

1. Adaptability
2. Aftermath
3. Anticipation
4. Battle Armor
5. Big Pecks
6. Chlorophyll
7. Compoundeyes
8. Cursed Body
9. Damp
10. Drizzle
11. Drought
12. Dry Skin
13. Early Bird
14. Effect Spore
15. Flame Body
16. Flash Fire
17. Frisk
18. Gluttony
19. Guts
20. Huge Power
21. Hustle
22. Hydration
23. Ice Body
24. Inner Focus
25. Insomnia
26. Intimidate
27. Iron Fist
28. Keen Eye
29. Leaf Guard
30. Lightningrod
31. Limber
32. Minus
33. Mold Breaker
34. Motor Drive
35. Natural Cure
36. No Guard
37. Oblivious
38. Overcoat
39. Own Tempo
40. Pickup
41. Plus
42. Poison Touch
43. Pressure
44. Reckless
45. Regenerator
46. Rivalry
47. Rock Head
48. Run Away
49. Sand Force
50. Sand Rush
51. Sand Veil
52. Sap Sipper
53. Scrappy
54. Serene Grace
55. Sheer Force
56. Shell Armor
57. Simple
58. Skill Link
59. Snow Cloak
60. Snow Warning
61. Solar Power
62. Soundproof
63. Static
64. Steadfast
65. Sturdy
66. Swarm
67. Swift Swim
68. Tangled Feet
69. Technician
70. Telepathy
71. Thick Fat
72. Unaware
73. Vital Spirit
74. Water Absorb
75. Water Veil
76. Weak Armor
77. White Smoke

Mega Evolution Only:

1. Aerilate

Signature Abilities:

1. Air Lock
2. Aura Break
3. Bad Dreams
4. Ball Fetch
5. Battery
6. Battle Bond
7. Beast Boost
8. Berserk
9. Color Change
10. Comatose
11. Corrosion
12. Cotton Down
13. Dancer
14. Dark Aura
15. Dauntless Shield
16. Dazzling
17. Defeatist
18. Delta Stream
19. Disguise
20. Desolate Land
21. Emergency Exit
22. Fairy Aura
23. Flare Boost
24. Flower Gift
25. Forecast
26. Full Metal Body
27. Gale Wings
28. Galvanize
29. Gooey
30. Gorilla Tactics
31. Grass Pelt
32. Gulp Missile
33. Hunger Switch
34. Ice Face
35. Ice Scales
36. Illusion
37. Imposter
38. Innards Out
39. Intrepid Sword
40. Libero
41. Liquid Voice
42. Long Reach
43. Merciless
44. Mimicry
45. Mirror Armor
46. Multiscale
47. Mummy
48. Neutralizing Gas
49. Normalize
50. Parental Bond
51. Pastel Veil
52. Perish Song
53. Power of Alchemy
54. Power Construct
55. Power Spot
56. Primordial Sea
57. Prism Armor
58. Propeller Tail
59. Punk Rock
60. Pure Power
61. Queenly Majesty
62. Receiver
63. Ripen
64. RKS System
65. Sand Spit
66. Schooling
67. Screen Cleaner
68. Shadow Shield
69. Shields Down
70. Slow Start
71. Soul-Heart
72. Stall
73. Stalwart
74. Stamina
75. Stance Change
76. Steam Engine
77. Steelworker
78. Steely Spirit
79. Surge Surfer
80. Tangling Hair
81. Teravolt
82. Toxic Boost
83. Triage
84. Turboblaze
85. Victory Star
86. Wandering Spirit
87. Water Bubble
88. Water Compaction
89. Wimp Out
90. Wonder Guard
91. Zen Mode

First Ability Only:

1. Flower Veil
2. Fluffy 
3. Fur Coat
4. Hyper Cutter
5. Iron Barbs
6. Levitate
7. Magma Armor
8. Mega Launcher
9. Refrigrate

First Ability and Hidden Ability only:

1. Blaze
2. Bulletproof
3. Clear Body
4. Contrary
5. Cute Charm
6. Defiant
7. Electric Surge
8. Grassy Surge
9. Healer
10. Honey Gather
11. Immunity
12. Justified
13. Light Metal
14. Magic Bounce
15. Magician
16. Marvel Scale
17. Misty Surge
18. Overgrow
19. Pixilate
20. Prankster
21. Psychic Surge
22. Rattled
23. Rough Skin
24. Shadow Tag
25. Speed Boost
26. Stakeout
27. Stench
28. Sweet Veil
29. Torrent
30. Truant
31. Volt Absorb
32. Wonder Skin

First Ability and Secondary Ability only:

1. Cheek Pouch
2. Download
3. Filter
4. Forewarn
5. Illuminate
6. Liquid Ooze
7. Magnet Pull
8. Poison Point
9. Sand Stream
10. Shed Skin
11. Shield Dust
12. Solid Rock
13. Sticky Hold
14. Strong Jaw
15. Suction Cups
16. Synchronize
17. Tough Claws
18. Trace

Secondary Ability Only:

1. Arena Trap
2. Heatproof

Secondary Ability and Hidden Ability only:

1. Anger Point
2. Cloud Nine
3. Competitive
4. Heavy Metal
5. Infiltrator
6. Klutz
7. Magic Guard
8. Moxie
9. Poison Heal
10. Quick Feet
11. Rain Dish
12. Slush Rush
13. Sniper
14. Storm Drain
15. Super Luck
16. Tinted Lens
17. Unnerve

Hidden Ability only:

1. Analytic
2. Aroma Veil
3. Friend Guard
4. Harvest
5. Moody
6. Multiscale
7. Pickpocket
8. Protean
9. Symbiosis

Edit (01.01.2020): Added Sword and Shield additions.