Warning! Spoilers to Superman and the Authority #2 below!
The latest issue of Superman and the Authority proves that in the future, the Fortress of Solitude's trophy room will be far more impressive than anything the Batcave has to offer. Superman's Artic home has always been both a tribute to his homeworld Krypton and a storage space/museum for some of the greatest moments in his long career. While many fans gravitate to Batcave's preservation of the legacy of the Dark Knight and his allies, Superman's collection will include not only significant moments in human history but also pop culture itself.
In Grant Morrison and Mikel Janin's Superman and the Authority, the future Superman is losing his powers but despite this, he wants to continue and save the world while he's still able. Although he's faded in the background for some time, he makes an appearance to save his former critic and powerful enemy Manchester Black from trigger-happy authorities. After saving Manchester's life, the Man of Steel reveals his situation and asks Manchester to put their differences aside to help him accomplish his mission. After begrudgingly helping Superman fend off an attack from Kryptonian criminals trying to escape the Phantom Zone, Manchester agrees to work with Superman by recruiting a new team of superheroes.
In Superman and the Authority #3, Superman continues to train and monitor his strength and speed with a workout regiment you won't see in the Batcave anytime soon. When Manchester meets with him, they walk through the Fortress of Solitude talking about their new potential recruits such as Natasha Irons, Midnighter, and Apollo. Superman's trophy room has surpassed being just memorabilia from lost civilizations and famous battles stored in one place. Symbols of human tragedies like the Titanic and the Space Shuttle Columbia are there, restored alongside what appears to be locomotives, winged alien vessels, wooden ships, and of course a classic Batmobile.
Superman has always been a chronicler of history, both human and alien, and even in old age, he hasn't lost his flair for sentimentality. Whether it's Grant Morrison or Mikel Janin's idea, they fill this single page with enough Easter eggs for any fan to appreciate. While the Titantic did appear in Morrison's All-Star Superman, the Time Machine from both the 1960 and 2002 version of The Time Machine, a red Tardis from BBC's Doctor Who and even the back end of the Delorean from the Back to the Future can be seen. The presence of these many iconic symbols of time travel could suggest an unconscious desire to turn back the clock, seeing that Superman laments about not keeping his promise to the late President John F. Kennedy when he had the power to do so. Although the Batcave boasts its own unique highlights amongst its gallery, Batman's dedication to his crusade against crime leaves little time to house any memorabilia outside of his famous identity while Superman's space and outlook allow him the time and ability for such elaborate tastes.
Although the Fortress of Solitude will surpass the Batcave, Morrison can't resist poking fun at Superman through Manchester Black who can't believe when he learns they'll be taking the Supermobile. Having already a Batmobile within his personal museum, Manchester mocks him for possibly having a little bit of Batman envy which Superman shrugs off. While Superman's recruitment will take him to all sorts of unlikely places, one would only hope Batman isn't alive to learn that Clark's Batcave was better in the end.