The best loseless format for saving pixel art is PNG. But as pixel art have very small size in most cases (like 32x32 or 64x64), most image viewers (like Gallery in Android or Photos in iOS) will enlarge it with blurring. That's how anti-aliasing works in the real world (unlike in pixel perfect world). Don't worry, your pixel art is not blurred in fact. Moreover, if you're gonna share it on social networks, Pixel Studio can upscale it when saving. For example, when saving 32x32 pixel art you can upscale it x16 to get good looking 512x512 image with clear pixel edges. Just select scale toggles available in Preview window. To share your pixel art on mobile platforms just open Menu/Share, this will open system sharing dialog with all possible options like Facebook, Twitter and other.
BTW, Pixel Studio saves all projects as PSP (Pixel Studio Format). This is internal format and such files can't be opened with other software. PSP, unlike other formats, stores your layers, frames and action history. So always use it if you are planning to continue your work.