Sorted the Challenges list by category. I copied the list itself from another website so I haven't checked the conditions for accuracy or typos.
1. Clear Classic on intensity 9.0 without losing a life: Double Final Smasher Protection Badge equipment
6. Clear Solo Classic in 12 minutes on intensity 9.0 with Marth: Critical Hitter Sword equipment
21. Clear Classic on intensity 9.0 without any customizations: Knuckle Joe trophy
23. Clear Classic on intensity 8.0 with 3 different characters: Master Fortress: Second Wave CD
33. Clear Classic on intensity 7.0 or higher with all characters: Regal Crown Mii headgear
46. Clear Classic on intensity 8.0 with 2 or more characters: Master Fortress: First Wave CD
72. Clear Solo Classic within 20 minutes with Olimar: Mission Mode CD
74: Clear Solo Classic at intensity 5.5 or higher with Wario, without any customizations: Full Steam Ahead (Spirit Tracks) CD
90. Clear Solo Classic on intensity 5,5 or higher with Kirby: Kirby's Adventure Masterpiece
98. Clear Solo Classic without losing a single life with Little Mac: Title (Punch-Out!!) CD
102. Clear Classic on intensity 5.5 or highter: Master Core CD
104. Clear Solo Classic with Yoshi: Yoshi Masterpiece
108. Clear Classic without losing a single life: Dr. Kawashima trophy
115. Clear Classic on intensity 2.0 or more with 5 characters: Master Hand trophy, Golden Hammer
2. Play as Lucina and score 8 or more KOs alone in Cruel Smash: Vampire Brawn Badge equipment
4. Play as Pit and score 4 or more KOs alone in Cruel Smash: Reaper trophy
9. Get 8 or more KOs in Solo Cruel Smash: Toadsworth trophy
39. Play Cruel Smash alone with Luigi and survive for 1 minute: Mr. L trophy
71: Get 1 or more KOs in Cruel Smash: Duck Hunt Stage and Character
3. Play as Dr Mario and get Fever Rush 8 or more times alone in Trophy Rush: 100,000G
12. Destroy a total of 50,000 blocks in Trophy Rush: Moon Launcher Protection Badge equipment
36. Play Samus and get a chain of at least 100 in Trophy Rush alone and without taking damage: KO Healer Arm Cannon equipment
50. Get a chain of 300 or more in Trophy Rush: CommanderVideo trophy
58. Play Fox alone in Trophy Rush and get Fever Rush 6 or more times in a single game: Krystal trophy
62. Get a chain of 200 or more with Donkey Kong in Trophy Rush: DK Rap CD
70. Destroy 250 blocks in a single game of Trophy Rush: Crazy Orders Pass
78. Play Robin alone in Trophy Rush and get Fever Rush 4 or more times: Conquest (Ablaze) CD
79. Play Pikachu alone in Trophy Rush and destroy a total of 200 blocks: Xerneas Pokeball
84. Play Falco alone and get a chain of at least 150 in Trophy Rush: Accele-Reflector custom move (Falco)
89. Play Mr. Game & Watch alone and destroy 100 blocks in a single game of Trophy Rush: Flat Zone X Stage
120. Get a chain of 100 or more in a single game of Trophy Rush: Chomp Hat Mii headgear
130. Get a chain of at least 50 in Trophy Rush while playing alone and as Wii Fit Trainer: Wii Fit U Trainer trophy
5. Play Pac-Man in Stage 3 of Solo Target Blast and score 300,000 using the back (left) wall: Whomp trophy
14. Score 450,000 in a single game of Target Blast Stage 3: Magolor trophy
17. Get a Perfect on every stage in Target Blast: 20,000g
35. Play Bowser Jr. in Solo Target Blast Stage 1 and score 150,000: Countdown Drill equipment
51. Play Ness in Stage 3 of Solo Target Blast and get 200,000: Jeff trophy
53: Get a Perfect on any stage in Target Blast: 5,000g
66. Score at least 200,000 in a single game of Target Blast Stage 2: Dire Hydrant custom move (Pac-Man)
73. Play as Zelda in Solo Target Blast Stage 1 and get 150,000: Wolf Link trophy
91. Play Palutena on Stage 2 of Solo Target Blast and score 80,000 or more using the back (left) wall: In the Space-Pirate Ship CD
100. Play Toon Link in Solo Target Blast Stage 2 and get 120,000: Lor Starcutter trophy
118. Score at least 150,000 in a single game of Target Blast Stage 1: Exploding Popgun custom move (Diddy Kong)
7. Collect all custom moves: Smooth Lander Brawn Badge equipment
8. Collect all Mii outfits: Able Sisters trophy
10. Collect all Mii headgear: Majora's Mask trophy
15. Collect over 300,000g: Prince of Sable trophy
16. Collect 700 unique trophies: Item Hitter Agility Badge equipment
20. Collect all CDs: Tac trophy
65. Collect 300 unique trophies: Timmy & Tommy trophy
69. View every fighter's Final Smash: Smash Ball trophy
116. Collect 50 different special moves: Nimble Dodger Agility Badge equipment
122. Collect 15 different special moves: Air Attack Brawn Badge equipment
125. Collect 50 unique trophies: Celest trophy
11. Clear All-Star on Hard with all characters: Saki Amamiya trophy
13. Clear True Solo All-Star without recovering health between rounds with Captain Falcon: Deathborn trophy
22. Clear True All-Star in 6 minutes with Jigglypuff: Koffing trophy
25. Clear True All-Star on Hard with Duck Hunt: Samus (Dark Suit) trophy
27. Clear True Solo All-Star Mode within 6 minutes with Shulk on any difficulty: Mechonis trophy
29. Clear True All-Star Mode on Normal or higher without using healing items: 150,000g
31. Clear True All-Star Mode without using healing items with Zero Suit Samus: Gunship trophy
37. Clear All-Star on Hard with 8 different characters: Caloric Immortal Protection Badge equipment
38. Clear True All-Star on Hard alone with Ike: Black Knight trophy
44. Clear True All-Star mode on Hard: Perfect-Shield Helper Brawn Badge equipment
75. Clear Solo All-Star on Normal or higher with Lucario: Meloetta Pokeball
109. Clear All-Star on normal or higher: Hyper Smasher Brawn Badge equipment
140. Clear All-Star: Victini Pokeball
18. Clear Crazy Orders after 20 or more turns with Mario: F.L.U.D.D. trophy
42. Clear Crazy Orders after 12 or more turns: Soda Popinski trophy
60. Clear Crazy Orders after 10 or more turns with Greninja: Battle! (Reshiram/Zekrom) CD
93. Clear Crazy Orders after 5 or more turns: Speed Crasher Agility Badge equipment
123. Clear Crazy Orders while playing as Villager: Rover trophy
128. Clear Crazy Orders: Crazy Hand trophy
41. Clear a Master Orders ticket on Intense: Desperate Specialist Agility Badge equipment
55. Clear a Master Order on Hard or higher with King Dedede: King Dedede's Theme CD
61. Clear a Master Order on Very Hard or higher: Trade-Off Attacker Protection Badge equipment
127. Clear a Master Orders ticker while playing as Mega Man: Tornado Hold custom move (Mega Man)
132. Clear a Master Orders ticker while playing as Rosalina & Luma: Rosalina in the Observatory/Luma's Theme CD
133. Clear a Master's Order ticket: Crazy Orders Pass
19. Score 85 or more in a single Rival Smash: Waluigi trophy
43. Get 3 or more KOs on your rival in Rival Smash: Dixie Kong trophy
59. KO your rival 4 or more times in Rival Smash: Ice Climbers trophy
94. Get a score of more than 20 in a single Rival Smash: Star Wolf (Star Fox: Assault) CD
96. KO your rival 2 or more times in Rival Smash: Super Mario Bros. 3 Medley CD
99. Obtain a score of 25 or more in a single Rival Smash with Dark Pit: Guiding Bow custom move (Dark Pit)
24. Start the final battle in Smash Tour with 10 fighters: Flying Man trophy
63. Get 3 or more Checkpoint Bonuses in Smash Tour: Bionis trophy
67. Defeat 12 or more fighters in a single Smash Tour: Crazy Orders Pass
68. Come in 1st in 5 or more Smash Tour battles: 2,000g
87. Use a Tour Item 18 times or more in Smash Tour: Riki trophy
92. Deal the final blow to a boss in Smash Tour: Ridley trophy
117. Recover the stat boosts a Metroid stole from you in Smash Tour: Vs. Parasite Queen CD
129. Play all the maps in Smash Tour: Pac-Land stage
134. Bump into an enemy who appears on the board during Smash Tour: Nabbit trophy
138. Bump into every opponent on the board in a single game of Smash Tour: Mecha Suit Mii outfit
26. Use Ganondorf in Home-Run contest and hit the Sandbag at least 3,280ft: King K. Rool trophy
40. Hit the Sandbag over 3,280 feet win Home-Run Contest: Daisy (Baseball) trophy
77. Hit the Sandbag between 1,640 and 1,656 ft. with ROB in Home-Rune Contest: Dr. Light trophy
82. Hit the Sandbag between 1,640 and 1,656 ft. in Home-Run Contest: Wonder-Green trophy
86. Hit the Sandbag 1,968 ft, or more in Home-Run Contest: Earthbound Masterpiece
95. Play Charizard and get 328 ft. or farther without using the bat in Home-Run Contest: Rock Hurl custom move (Charizard)
106. Play Peach and hit the Sandbag 1,049 ft. in Home-Run Contest: Title/Ending (Super Mario World) CD
28. Clear 3-Minute Smash after getting 120 KOs: Kritter trophy
49. Clear Solo 3-Minute Smash with 100 KOs with Mii Gunner: Birdo trophy
56. Clear 3-Minute Smash with 110 Kos with Boswer: Banzai Bill trophy
113. Clear 3-Minutes Smash after getting 60 or more KOs: 1,000g
30. Complete the reward conditions on all Solo Events: Galacta knight trophy
32. Clear the "All-Star Battle: Secret" Event on Hard: Shadow trophy
34. Clear the "The FINAL Battle: Event on Hard: Soulflee trophy
47. Clear "Kirby's Crazy Appetite" Event on Hard: Fire Kirby trophy
54. Clear all Solo Events: Special Flag trophy, Golden Hammer
83. Clear the "Fitness Junkie" Event on Normal or higher: Snorlax trophy
97. Clear the "No Mere Sparring Match" Event on Normal or higher: Punch-Out!! Featuring Mr. Dream Masterpiece
111. Clear the "Behind Enemy Lines" Event: Tricky trophy
119. Clear the "The Falchion;s Seal" Event on normal or higher: Dashing Assault custom move (Marth)
124. Clear the "When Lightning Strikes" event: Pokemon Stadium 2 Stage
126. Clear the "Playing Tricks" event: Smashville stage
139. Clear the "The Original Heavyweights" event: Kongo Jungle 64 stage
45. Get 120 or more KOs in a single Endless Smash: Unharmed Attacker Agility Badge equipment
52. Play Meta Knight and score 90 KOs in Solo Endless Smash: Substitute Doll trophy
81. Play Sheik and score 50 KOs or more in Solo Endless Smash: Tetra trophy
88. Play Link and score 50 KOs in Solo Endless Smash: Ripping Boomerang custome move (Link)
107. Get 50 or more KOs in a single Endless Smash: No-Flinch Smash Protection Badge equipment
48. Clear 10-Man Smash within 25 seconds with all characters: Wonder-Yellow trophy
57. Clear 10-Man Smash alone with Dr. Mario without taking any damage: Dr. Mario Masterpiece
80. Clear 10-Man Smash in 17 seconds: Gerudo Valley CD
105. Clear 10-Man Smash in 25 seconds: Fancy Suit Mii outfit
121. Clear Solo 10-Man Smash within 35 seconds while playing as Mii Brawler: Fighter Uniform Mii outfit
131. Clear 10-Man Smash: 100G
64. Clear Solo 100-Man Smash in 3 minutes with Mii Sword-fighter: Multi-Man Melee 2 (Melee) CD
76: Clear Solo 100-Man Smash in 3 minutes with Sonic: Speed Skater Shoes equipment
110. Clear 100-Man Smash in 3 minutes: Wii Fit Plus Medley CD
112. Clear 100-Man Smash: Tatris: Type B CD, Golden Hammer
85. Get a max combo of 50 or more in Training mode: Unharmed Speed Demon Agility Badge equipment
103. Get a max combo of 30 or more in Training: Title Theme (Wii Sports Resort) CD
101. Reveal the entire wall during the credits: Credits CD
114: Have the game on for more than 10 hours: Mr. Resetti trophy
135. Create a Mii Fighter in Custom: Peach Crown Mii headgear
136. Customize a fighter in Custom: Shield Regenerator Protection Badge equipment
137. Create a stage in Stage Builder: Saharah trophy