This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to mix slowly. For example, does the standard chain on the hardcore model mix extremely slowly on any interesting graphs? Are there bad row/column sums for which the Diaconis/Sturmfels design walk mixes slowly? Are there interesting families of graphs for which the Broder/Aldous algorithm gets stuck?
I apologize that there is obviously not going to be a single 'right answer' here. I need some pathological examples, and don't know of any that are considered interesting these days.