David Bellamy constructed a surprising example of a smooth dendroid in the plane with a connected set of endpoints. In this talk, I will present the new result that any planable smooth dendroid with $1$-dimensional endpoint set must contain a Cantor fence (a copy of $2^\omega \times [0,1]$) or a Bellamy dendroid (a smooth dendroid whose endpoint set is connected). This is false outside the plane, and it is unknown whether every Bellamy dendroid contains a Cantor fence. More generally, a continuum is said to be non-Suslinian if it contains an uncountable family of pairwise disjoint, non-degenerate subcontinua. I will discuss some open problems about this property in Julia sets and other plane continua with rich dynamical structures. Among these are: If a plane continuum admits a mixing homeomorphism, then is it non-Suslinian? Is the Sierpiński carpet the only locally connected plane continuum that admits a mixing homeomorphism?