Superheroes have their headquarters, but so do supervillains! Of course, trying to conquer the world takes some time, and so the supervillian needs a place that not only is conducive to it, but also allows him to relax and play air hockey in-between. Or any other game that helps him keep up his mood for world conquest. Of course, the place can reflect something of the villain's personality, which is important of course. And yes, don't forget, it probably has a torture chamber or a dungeon for imprisoning or torturing the villain's dreaded enemy.
The Death Star
The most ominous and among the largest, the Death Star is considered the very embodiment of galactic terror. It content among the most badass evil guys in space and can blow up a planet. Although they should do something about vulnerable vents. And one wonders where they got enough materials to build it.
Lex Luthor, Superman's nemesis, makes up for his lack of super-powers with his being a brilliant scientist and businessman. And even president of the United States! Of course, Luthor does it from his LexCorp headquarters, the cartoon version featured here.
The evil counterpart to Castle GreySkull, Snake Mountain is one heck of a lair. That giant snake winding around giant rocks leaves you no doubt, it's an evil guy's place. You even wonder if it'll come to life to eat up He-Man like Thor vs. Jormungandr.
Among the supervillains taking to space for refuge, Magneto has this neat asteroid retreat where he hid away for a while until some psychotic wacko convinced to get back to villainy. These days however, he helps out the X-Men and other mutants. But for a time, he did seem unreachable in space.