<p>The link over here lists ([http://www.yoda.arachsys.com/csharp/singleton.html][1]) some singleton patterns in C#. The article also describes the obvious that a singleton is not meant to accept parameters which “as otherwise a second request for an instance but with a different parameter could be problematic”. This means that any parameters you need to get the class working should be induced as a property.  </p>

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I am curious to know if there are any parameterized singleton design patterns out there. Accepting values as a property does not enforce anything to the consumer.
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Based on your question, it seems you may be looking at an Abstract Factory pattern (creates an instance of several families of classes) that keeps an internal list/dictionary of classes that have already been instantiated, thus mimicking the singleton pattern functionality.
You would then use this factory class to request an object based on parameters you've passed in, and if it exists in its internal list it gets returned, and if not, a new instance is created and then added to the list and returned.