There are some wonderful and well known applications of the Baire Category theorem that prove the existence of certain objects without constructing them, e.g. there exists a continuous nowhere differentiable function (in fact the set of functions differentiable somewhere is meager). In the same spirit, I'd like to if and what the Baire Category Theorem can say about connected sets in $\mathbb{R}^n$ (and the like).
Question 1: Can we prove that their exists a connected set that is not path-connected using the Baire category theorem? If so, is the set of path-connected sets meager in the space of connected sets?
Question 2: If we have success with Question 1, why not go crazy and ask about locally connected sets in the space of (path-)connected sets, or locally path-connected sets in the space of locally connected (path-)connected sets?
I've tried to answer the above questions myself trying to mimic the proof of the existence of a continuous but nowhere differentiable function but I don't really know where to begin.