Examples of continua that are contractible but are not locally connected at any point A continuum is a compact, connected, metrizable space.
What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of connected sets?
The following is an example of a contractible nowhere locally connected metrizable space, but I'm not aware of any compact examples (every segment on the "main line" has segments sticking out on a dense set).

This question was previously asked on MSE but didn't get an answer there.
 A: An example of such a space was constructed by Edwards in "R. Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S. 20 1999), 494.". The example is the following space:
Start with the space $X$ obtained stacking a sequence of Cantor fans (cones over the Cantor set) end to end as depicted.

Then consider the space $X\times\Bbb R$, which looks like the upper half-plane with a lot of "fins" attached to it. Let $X'$ be the one point compactification of $X\times\Bbb R$, so that it looks like a closed disk with a lot of circular fins attached. The space $X'$ is a contractible continuum and it is locally connected precisely on the boundary of the circle. To make it nowhere locally connected construct $Y$ from $X'$ by wrapping one more Cantor fan around the circle, the resulting space is still contractible as explained here and clearly nowhere locally connected (in fact it is also nowhere connected im kleinen).
Now I'm wondering whether a planar contractible continuum must be locally connected somewhere, but that should be a separate question.
