When should I use a semicolon after curly braces?

You use a semicolon after a statement. This is a statement:

var foo = function() {
  alert("bar");
};

because it is a variable assignment (i.e. creating and assigning an anonymous function to a variable).

The two things that spring to mind that aren't statements are function declarations:

function foo() {
  alert("bar");
}

and blocks:

{
  alert("foo");
}

Note: that same block construct without semi-colon also applies to for, do and while loops.

It matters too when you intend to minify your code.

So I personally add one after every } where automatic semicolon insertion (ASI) would insert one.

I wrote a post about ASI in JavaScript.

Don't use a semicolon:

...if it's just your every-day function declaration:

function foo() {

} // No semicolon

Use a semicolon:

...if it's an assignment:

var foo = function() {

}; // Semicolon

...or a self invoking function:

(function () {

})(); // Semicolon

You never need to; you always can (except before else and while).

Explanation:

Unfortunately, JavaScript semicolons are optional.
Therefore, you never need to add a semicolon.

It is (very) good practice to terminate every statement with a semicolon.
The only statements that end with a } are statements ending with an object literal (e.g. JSON) or function expression.

Therefore, best practice is to put semicolons after the following two braces (only):

var myFunc = function() { };
var myobject = { };

If we have a self-invoking function, we need to put a semicolon before it, otherwise it becomes part of the previous assignment statement. Consider the following:

testClass = function(name) {
  document.write ("Instantiating testClass<br />");
  this.name = name;
}

testClass.prototype.report = function() {
  document.write ("I'm " + this.name + "<br />");
  return 1;
}

testClass.prototype.testMethod = function(param) {
  document.write ("Running testMethod with parameter value " + param + "<br />");
  return 2;
} // notice that there is no semicolon here

(function() {
  document.write ("Running self-invoking function<br />");
  return 3;
}());

if (typeof(testClass.prototype.testMethod) !== "function") {
  document.write ("testMethod type: " + typeof(testClass.prototype.testMethod));
  document.write (", value: " + testClass.prototype.testMethod + "<br />");
}
var testOb = new testClass("Bill");
testOb.report();
testOb.testMethod(4);

This will produce the following output:

"Running self-invoking function
Running testMethod with parameter value 3
testMethod type: number, value: 2
Instantiating testClass
I'm Bill"

...plus a JavaScript error reported by the browser: testOb.testMethod is not a function

This is certainly not what we intended. Why is testMethod running immediately, before we have even instantiated the class? And why does it no longer exist when we want to call it as a member method?

What is happening is that testMethod is being assigned not our function definition, but the return value of the function definition. And the function definition itself is being run anonymously. This is how:

1. The testClass constructor and the member method report are successfully defined/assigned.
2. Because of the lack of a semicolon after the definition for testMethod, the () surrounding the following self-invoking function becomes an invocation operator, which causes what we think is our definition of testMethod to become an anonymous function that is invoked immediately, and the return value of the following anonymous function becomes its parameter list. This explains the order of printed output - our self-invoking function is run first as it is evaluated as a parameter.
3. Since our intended function definition returns 2, it is this 2 that is assigned to testMethod, and not the function definition. This is confirmed by our printing of the type and value of testMethod.
4. Now testClass is successfully instantiated as testOb and its report method works as intended, proving that the class definition is otherwise intact.
5. When we try to call testMethod, we are told by the interpreter that it is not a function - and rightly so, because it is a number with the value 2.
   

If we put a semicolon after the definition of testMethod, it will separate its assignment from the calling of the self-invoking function, and we will have the result we expected:

"Running self-invoking function
Instantiating testClass
I'm Bill
Running testMethod with parameter value 4"

Or we could even put it directly before the anonymous function:

;(function() {...

But I suggest that since the problem is due to the lack of a semicolon at the end of an assignment statement, we should perhaps make a habit of always putting a semicolon after defining functions in this way. i.e. all of my functions above should have a semicolon after the closing brace, because they are all assignments of anonymous functions.