Geography of the Teichmüller stack

In this article, we describe the geography of the Teichmüller stack of \cite{LMStacks} and of one of its variants we introduce here, giving some answers to questions as: which points are orbifold points? What are the different local models of special points?.. We give a rough description in the general case, and we use the compacity of the cycle spaces to get a much more detailed picture in the Kähler setting.