A superhero needs a place to stay. Of course, it doesn't need to be only their apartment when in their civilian identity. It can be a high-profile base, facility or building which indicates to the public that the hero is available to help them. Or it could be as secretive as the Batcave, where Batman keep all his gear, vehicles and equipment, and do his detective work without interference. Superhero lairs are functional, not just because they are a place to stay, but contain all the tools of the hero's trade.
Perhaps the most famous superhero lair, the batcave is a secret to most of his enemies. Located under the Wayne manor in most stories, it is perhaps the most memorable of superhero lairs. It's got everything from his crimefighting computer, equipment storage and his vehicle docks to a mini-museum. And it so far remains secret even to Batman's worst enemies.
Fortress of Solitude
Superman's fortress, located in the North Pole, is yet another recognizable lair. Perhaps the most popular version in the crystal-themed fortress in the Christopher Reeve version of Superman. He not only keeps Kryptonian technology in the fortress, but spoils from defeated enemies as well.
The Fantastic Four's headquarters and where Reed does his experiments, as well as store the team's equipment and vehicles. Of course, there's the lounge where he interacts with his family. And where he handles his business... I wonder if he had any businesses at all.
The Avengers' headquarters was formerly a mansion owned by Tony Stark. Jarvis worked here as the team's butler. Oh, and Jarvis is a human, not a computer.