Young Justice, in part, is what gave me the idea for a home base with the occasional away mission for the reasons stated above (though to be fair, my only experience with that series was the show on Cartoon Network, which didn't really go into great detail about the area around their base). The other reason I suggested that idea is that it is very flexible, it can be sustained indefinitely or done away with as the plot demands, while also adding some stability to the setting.
Of course, I do generally tend to like Superhero RPs like this to have a more small team centered feel to them, which is part of the reason I'm against adding villainous PCs. Though this is a preference I developed after experiencing quite a few RPs succumb to in-fighting and other such unpleasantness. It can also be attributed to a lesser extent to my long standing love of shows such as Justice League,Young Justice, and (to a slightly lesser extent) Teen Titans, where there is a clear focus on one side/team, and while it does deal with their opposition in a more person light from time to time, the focus never clearly leaves the main team.
Edit: I found a lot of inspiration in the DCU, specifically Justice League and Young Justice. Also, while it's not exactly superhero related, I get some inspiration from some manga I read and the music I listen to, specifically the lyrics and how I interpret them (even if I know the intended meaning of the song writer). I enjoy Rock/Alternative Rock for this purpose. I like the idea of a close group of young superheroes working together from Young Justice (I believe they were all teenagers in that, however, I favor an age range of about 18-25).
I love the dynamic of the Justice League too. All these great and powerful heroes, who can work very well on their own, but have such an incredible community that they can call on for help if/when they need it.
The idea of mentor heroes letting their proteges work on their own missions (granted, with a whole team of other proteges) in order to both further their training and give them the freedom of discovery to really understand what drives them, and allows them to step out of the shadow of their teacher inspires my writing as well.
And though it's not related to the superhero genre specifically, I absolutely adore the ability to put my character in a world with my fellow RPers and the GM in order to see how my character will both affect the world around them, and how they'll in turn change because of that same world.
Those are some of the things that inspire me to write in Superhero RPs, and in some cases, RPs in general.