If it's speech then I say "Super Nintendo," and might tack on the "Entertainment System" if it's someone unfamiliar with it. In writing, I've done them all from time to time, depending on the context.
In the form of answering a question, I'd say the full name, "Super Nintendo Entertainment System" In casual conversation, "snes", "Super N.E.S.," or "S.N.E.S." Rarely, "The Super" Occasionally I'll use "Super Famicom" or its abbreviations when referring to the Japanese versions of games.