As others have mentioned the dollar sign is intended to be used by mechanically generated code. However, that convention has been broken by some wildly popular JavaScript libraries. JQuery, Prototype and MS AJAX (AKA Atlas) all use this character in their identifiers (or as an entire identifier).
In short you can use the $ whenever you want. (The interpreter won't complain.) The question is when do you want to use it?
I personally do not use it, but I think its use is valid. I think MS AJAX uses it to signify that a function is an alias for some more verbose call.
For example:
var $get = function(id) { return document.getElementById(id); }
That seems like a reasonable convention.