How static variable can retain incremented value

That isn't your code: I can tell for two reasons.
1) It won't compile - C# is case sensitive, and Public is not the same as public
2) Your Singleton class as shown doesn't contain a getInstanse method, which is crucial to your problem.

The whole idea of a singleton class is that there is one and only one instance of the class in the system - so the constructor is always declared as private, not public so it can only be created inside the class itself. Normally that would be in a GetInstance method or similar, but the instance would be retained in a private static variable inside the class:

class Singleton
    {
    private static int counter = 0;
    private static Singleton instance = null;
    private Singleton()
        {
        counter++;
        Console.WriteLine("Counter-" + counter);
        }
    public void getMessage(string s)
        {
        Console.WriteLine(s);
        }
    public static Singleton GetInstance()
        {
        if (instance == null) instance = new Singleton();
        return instance;
        }
    }

class Program
    {
    static void Main(string[] args)
        {
        Singleton objSingleton = Singleton.GetInstance();
        objSingleton.getMessage("Hi This is my first message!");
        Singleton objSingleton2 = Singleton.GetInstance();
        objSingleton.getMessage("Hi This is my second message!");

        Console.ReadLine();
        }
    }

And that's why you only ever get a counter of 1: only one instance is ever created.