Adding a static object to a resource dictionary

Unfortunately it is not possible from XAML. But you can add the singleton object to the resources from code-behind:

public partial class App : Application
{
    protected override void OnStartup(StartupEventArgs e) {
        base.OnStartup(e);

        Resources.Add("MySingleton", Singleton.Instance);
    }
}

It is possible in XAML:

<!-- assuming the 'my' namespace contains your singleton -->
<Application.Resources>
   <x:StaticExtension Member="my:Singleton.Instance" x:Key="MySingleton"/>
</Application.Resources>