The silver age Superman had bases on an undersea cliff of Earth-One as well as an asteroid in space before finally settling on his Fortress of Solitude in the Arctic. Accessible by a gigantic golden key, it housed various chambers with memorabilia of Krypton and its owners exploits as well as family.
The golden age Superman initially had his residence in mountains outside Metropolis of Earth-Two, known as the Secret Citadel. While he had other locales such as an Arctic palace in the late 1940's and an attempt to replicate Superman's own Fortress of Solitude, he settled into his permanent home back at the Citadel that began it all!
Of course, Supergirl had her own Fortress in a desert which she soon after disassembled. And, lest we forget, Krypto the Superdog had his own Doghouse of Steel... most likely converted from Kal-El's original asteroid headquarters.
Of interest is the fact that deep under the surface of Earth-Two was the Ultra-Humanite's subterranean headquarters, much like the under-the-sea Fortress of Earth-One, from which Ultra accessed his Superman's Secret Citadel. And this Citadel was located very close to if not in the same location as Brainwave's hidden laboratory, within a mountain outside of Earth-One's Metropolis! This lab was used as an access point to Earth-Two's Sinister Citadel, home of Brainwave's Secret Society of Supervillain allies.