There is a theorem by Moore that says there are not uncountably many disjoint copies of the simple triod in the plane (the simple triod is the space by adjoining one end point from three copies of $[0,1]$ together to make a three-pointed asterisk). I am wondering what other compact, connected spaces also satisfy this.
More specifically, for any compact subset of the plane there always exist countably many disjoint copies (easy to see taking disjoint discs and the fact that they are homeomorphic to the plane), but which ones don't have uncountable such collections? Of course any space must be one-dimensional, except the example of a singleton, and none can contain a simple triod.
For locally connected, compact, connected subsets of the plane, if they are not an arc or a circle then they contain a simple triod, by some standard theorems (they are locally path-connected), so examples need to be a bit gross. Just by intuition it seems like the sin$(\frac{1}{x})$ continuum (closed topologist's sine curve) does not work, maybe because it has some similarity to a triod.
There are some results using the concepts of span, but I am wondering if there are any other sort of . . . sporadic examples that are easy to see naturally.