// класс, который должен быть одиночкой

 public class DataStore
    {
        public string Name { get; set; }
    }

Kernel.Bind<DataStore>().ToSelf().InSingletonScope();
//thread-safe Tests

         (new System.Threading.Thread(() =>
            {
                DataLayer.DataStore r1 = ioc.Ioc.Get<DataLayer.DataStore>();
                r1.Name = "From BackgroundThread";
                Debug.WriteLine(r1.Name);
            })).Start();

            DataLayer.DataStore r = ioc.Ioc.Get<DataLayer.DataStore>();
            r.Name = "FromMainthread";
            Debug.WriteLine(r.Name);
//output
//FromMainthread
//FromMainthread
/////////////////////////////////

// те же самые тесты
 Kernel.Bind<Lazy<DataStore>>().ToConstant(new Lazy<DataStore>(
() => new DataStore()
)).InThreadScope();

//From BackgroundThread
//Поток 0x3714 завершился с кодом 0 (0x0).
//FromMainthread

/////////////////////////////////

 Kernel.Bind<DataStore>().ToSelf().InSingletonScope();

//FromMainthread
//FromMainthread

public class Singleton
{
    private static readonly Singleton instance = new Singleton();
 
    public string Name { get; private set; }
 
    private Singleton()
    {
        Name = System.Guid.NewGuid().ToString();
    }
 
    public static Singleton GetInstance()
    {
        return instance;
    }
}

 public class DataStore
    {
public ObservableCollection<FakeClass> FakeList { get; set; }
    }