Designing Singletons in Node.js
fredkschott.com
fredkschott.com
That said, it's not necessarily required. You can make any of your module files export only one function, a function that defines a class. This is the preferred pattern in most programming languages. I then use dependency injection to propagate my state around.
And that's generally the case in any programming language: if you're thinking of a singleton, dependency injection is the answer.
BTW, do you write javascript to create apps on mobile?