Unlock Jigglypuff (aka. Purin) as a fighter
Complete Classic Mode or Adventure Mode with any character on any difficulty or stock. Once you have achieved this, Jigglypuff will challenge you. Beat him to unlock him.
Unlock Luigi as a fighter
Play Adventure Mode, and in the first stage (the Mushroom Kingdom part) hit the flag at the end with the last digit of the seconds-counter being a "2" (eg. 03:32:56). When you fight at Peach's Castle in the next part of the stage, you will battle Luigi instead of Mario. Continue as normal until you finish Adventure Mode. Once you have achieved this, Luigi will challenge you. Beat him to unlock him.
Unlock Mewtwo as a fighter
EITHER play VS. Mode for a total of 20 hours overall (using more human players reduces the time needed) OR play 700 VS. Mode matches. Once you have achieved this, Mewtwo will challenge you. Beat him to unlock him.
Unlock Marth as a fighter
Use all 14 of the non-hidden characters at least once in either VS. Mode, Classic Mode, or Adventure Mode. Once you have achieved this, Marth will challenge you. Beat him to unlock him.
Unlock Mr. Game & Watch as a fighter
Complete Classic Mode, Adventure Mode, or Break the Targets with all 24 other characters. Once you have achieved this, Mr. Game & Watch will challenge you. Beat him to unlock him.
Unlock Dr. Mario as a Mario-clone fighter
EITHER clear Classic or Adventure Mode with Mario without using any continues (any difficulty or stock) OR play 100 VS. Mode battles. Once you have achieved this, Dr. Mario will challenge you. Beat him to unlock him.
Unlock Ganondorf as a Cpn. Falcon-clone fighter
Complete Event 29 "Triforce Gathering." Once you have achieved this, Ganondorf will challenge you. Beat him to unlock him.
Unlock Falco as a Fox-clone fighter
EITHER clear the 100 Man Melee OR play 300 VS. Mode battles. Once you have achieved this, Falco will challenge you. Beat him to unlock him.
Unlock Young Link as a Link-clone fighter
EITHER complete Classic Mode as Link, Zelda and 8 other characters, OR play 500 VS. Mode battles. Once you have achieved this, Young Link will challenge you. Beat him to unlock him.
Unlock Pichu as a Pikachu-clone fighter
EITHER complete Event 37 "Legendary Pokémon" OR play 200 VS. Mode battles. Once you have achieved this, Pichu will challenge you. Beat him to unlock him.
Unlock Roy as a Marth-clone fighter
Complete Classic Mode or Adventure Mode using Marth without continuing. Once you have achieved this, Roy will challenge you. Beat him to unlock him.
Unlock All-Star Mode
Unlock all 25 secret characters.
Unlock "Brinstar Depths" stage
Play 50 VS. Mode battles.
Unlock "Fourside" stage
Play 100 VS. Mode battles.
Unlock "Big Blue" stage
Play 150 VS. Mode battles.
Unlock Pokémon "Poké Floats" stage
Play 200 VS. Mode battles.
Unlock "Mushroom Kingdom II (Subcon)" stage
Get the Birdo trophy from the Lottery.
Unlock "Flat Zone" stage
Complete Classic Mode with Mr. Game & Watch.
Thanks to PikachuRoo for sending me that one.
Unlock "Battlefield" stage
Complete All-Star Mode with any character.
Unlock Master Hand's "Final Destination" stage
Complete all 51 Event matches.
Unlock Original SSB "Dreamland" stage
Complete Break The Targets with all 25 characters.
Unlock Original SSB "Yoshi's Island" stage
Hit the Sandbag over 1300ft in the HomeRun Contest.
Unlock Original SSB "Congo Jungle" stage
Complete the 15-Minute Melee.
Unlock Events 11-30
Complete the first ten events to open a few more. Keep completing events to open up to Event 30.
Unlock Events 31-39
Unlock Luigi and Falco as secret characters.
Unlock Event 40-50
Unlock all 11 secret characters.
Unlock Event 51
Complete Events 1 to 50.
Unlock Sount Test
Complete Event 51 and open all 29 stages.
Unlock Random Stage Switch
Unlock The five main secret stages, and Flat Zone.
Unlock Score Display
Achieve 5,000 KOs with a particular character (any character will do).
Bowser x1,000,000= Giga Bowser