Where to attach global scripts in Unity?

To expand on Tetrad's answer, consider a script like this:

public class GameSystem : MonoBehavior
{
  private GameSystem m_Instance;
  public GameSystem Instance { get { return m_Instance; } }

  void Awake()
  {
    m_Instance = this;
  }

  void OnDestroy()
  {
    m_Instance = null;
  }

  void Update()
  {
    // global game update logic goes here
  }

  void OnGui()
  {
    // common GUI code goes here
  }

  // etc.
}

You can then create an object called "GameSystem" in the root of your scene. The only components it would have would be the built-in transform component (set its position to the origin, its rotation to identity, and its scale to one; not that it matters, but it's good practice). Attached the GameSystem component to that object.

You can now access your global object by simply using GameSystem.Instance.blah(). Its event handler methods are invoked automatically by Unity since it derives from MonoBehavior and exists as a component. You can add fields to it that reference other game objects or components and connect them in the Unity object hierarchy view.

Yes, this is all a little "weird." It can feel a little dirty to have to create an object (that even has a transform) which is always just a global singleton. It's what Unity requires to create global objects that get an Update message and are manipulable with the default Unity property editor, though.