This post includes some mild SPOILERS for anybody who ever wants to play one of these, though I haven't run one in a couple of years now.
OK, the first All-stars game was Pulp, and the heroes were assembled when the dreaded Volcano Master set off a small volcano in the middle of Central Park. After dealing with that, the heroes track back the effects of the volcano machine, which leads them to the caldera of the volcano in the center of Skull Island only a couple of years after Kong's capture. They fight their way through dinosaurs and giant insects to find the Volcano Master, who is in fact in league with the Nazis in his plan to hold cities around the world for ransom.
2005 was 70s-80s heroes. When Steve Austin goes missing on a case, OSI calls FBI agent Bill Maxwell (from Greatest American Hero) to assemble a team of super-agents to track him down. (Maxwell is revealed early on to also be Kelly Robinson from I Spy, which is the one hint we allow.) The team picks up the trail of Los Angeles' master criminal Mister K, who uses amazing androids like Bigfoot and the KISS robots to commit crimes, and follow it all the way back to Fantasy Island (usually by taking the Love Boat, of course.) There, Mister Roarke and Tattoo are revealed to be the heads of the crime ring; more, Tattoo is actually Knickknack from Man With The Golden Gun, and Roarke is of course Khan Noonien Singh, using the advanced-robot-run island of fantasies to raise funds for the coming Eugenics War!
2006 was Universal Monsters. A mysterious benefactor gathers the players and offers to grant them unimaginable power if they can only recover a magical amulet from a mad scientist's island laboratory near Goldstadt, which is defended by sharks, mole men and a giant tarantula. Of course, none of the monsters wants the others to get the amulet, and nearly every time this was run the group broke down into complete interparty battle. The benefactor is revealed to be Lucy Weston, since unlike most versions of Dracula she escapes in the Universal movie.
2007 was 60s super spies. This time, it's James Bond who disappeared on a mission, and the spies are assembled to find him. They track clues that eventually lead to Goldfinger, who having survived his apparent death is auctioning the right to kill Bond off to an island full of insane villains. The plot also involves the acquisition of the secrets to Bionics, which are of course descended from the work of Victor Frankenstein (and Jaws is himself the first actual Bionic Man, with his deadly mechanical mouth!)
2008 was victorian heroes. An all-star team of villains have discovered Narnia, and are using the Wood Between The Worlds to travel to and rob an entire multiverse. This one was pretty nuts, even by my standards. I had fun with Alice being a slayer in Wonderland, though.
2010 pitted the actual heroes of KISS, along with Alice Cooper and Wendy O Williams, against a plot to steal the soul of a teen fan of the band. They traveled through the worlds of multiple concept albums (including the Yellow Submarine, Xanadu and others) and received assistance from various other bands (ZZ Top showed up to give them the keys to a car when needed, for example.) In the end, they learn that the Devil (played by Ozzy Osbourne) has been tricked into taking an innocent soul by Pat Boone, who hates that heavy metal has made Hell cool and wants to convert it to Easy Listening as the new ruler. The only way out for our heroes is to get the attention of a currently absentee God, who is played by Johnny Cash.