Target Test
This is the same as the target test in the Classic Mode. The stages are specifically designed for each character; no character can play another's level.
Target Test trials end when either every target is destroyed or the character falls off the zone and is KO'd.
Beating Target Test with all 25 characters (Sheik not included) will unlock Past Stages:Dream Land.
Home-Run Contest
You basically are on a stage with a sandbag and a home-run bat. Your job is to do the most damage to the sandbag without knocking it off the platform in 10 seconds and then whacking it with the homerun bat smash. It then tells you the distance you smashed it. All characters have the potential to hit the sandbag over 3000 feet, while Ganondorf and the Ice Climbers both have the potential to exceed the track limit (11,347.2 feet)
Knocking the sandbag around 1000 ft. will earn you the Sandbag trophy. If you get more than 1350 ft with Yoshi, then you get Past Stages:Yoshi's Island.
Multi-Man Melee
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10-Man Melee
Real easy stuff. You should have no problem beating this, however, getting a good time (under 10 seconds) on this is challenging for some characters.
100-Man Melee
100 kills may not sound that hard of these wimpy wireframes, but the weak punches will add up, and you may have over 200% damage when you finish this. Use your best character and try not to let them attack you or even get near you. If you finish you will be able to unlock Falco.
3 Minute Melee
See how many wireframes you can KO in 3 minutes. This is tougher than it sounds, however, because they start to wear you down after 2 minutes. Your best bet would be to avoid the two semi-pacifist wireframes in the beginning for about a minute, and then start slashing away at them. Or if you're good with your character, try to KO as many as possible.
15 Minute Melee
You must survive the wireframes for 15 minutes. Using either Jigglypuff or Kirby's ability to jump 5 times in a row it is possible to jump from one side of the stage to the other without being attacked. Or, you can do it the easy way, using DK's 
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move where he slaps the ground, it KOs the wireframes instantly. If you get hit, move to the other side and continue. You should beat this without getting over 100% damage. When beating this, you unlock Past Stages: Kongo Jungle.
Endless Melee
This is the most fun of the melee events because there is no set task to accomplish except beat your old score. KO as many of the wimpy wireframes as you can either by traditional means, or the cheap DK handslap method. You can score over 500 KOs easily with DK using this method.
Cruel Melee
The wireframes have been set to level 9 and aren't as light before (but still about as light as Mr. Game and Watch), and they seem to have bought a bunch of Judgment hammers at discount because sometimes one single attack from them can KO you at 0% damage, while they require being over 50% damage to KO normally. Your best bet is Roy with his 
+ Blazer move. Another good person to use is Jigglypuff with her rest attack. Just fly off the stage for a second and all of the wireframe people will come to the side you are on. If you do it right, you can kill of all five at the same time. After a few tries scoring KOs of 5 or more should be no problem. When (or if) you get 5 or more KOs in Cruel Melee, you'll unlock the Mr. Resetti trophy. If you get really good, you can continue playing this mode infinitely. It's like endless melee set to insane difficulty.