If you have long been programming in C++/COM and then you move to C#.NET, the first difference you can feel is that you got a ctor for the object you create unlike the CoCreateInstance. In the C++/COM world, you generally would have a Initialize method to do the constrcution sort of, paired with Terminate/Uninitialize method. Similar is the case with singleton classes. For singleton classes in C++, you will have public static Instance or GetInstance method to get the only and one instance of the class and then use the initialize method to do the construction. This is certainly advantageous than the ctor facility in .NET, since you will not know when the instance will be initialized without the initialze method. Any call like SingletonClass.GetInstance().SomeMethod may initialize the singleton anywhere and you will not exactly do the initialization during the application startup, which in many cases will lead to application errors after startup.
I do not recommend putting the initialization logic in the ctor, particularly for singletons. The Initialize/Uninitialize method seem to be primitive and kind of from the legacy age but we want code elegance rather than fashion. The pair gives a reasonable intuition and a feel of responsibility to initialize and uninitiailze. Without this simple pair, the object [singleton or any .NET object] gets initialized without control. Also the developers as soon as they enter the .NET world, with the advice from somebody next door, instantly or deliberately forget the memory management and leave everything to GC. But GC can perform the uninitialization required by the business logic.
The Initialize/Uninitialize pair just silently enforces to follow the pattern to initialize at the right place, and most important uninitialize, not giving the risk of remembering about Dispose.
I was forced to write this comment because I was forced to write that code.