Question: I would like to know which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. Thanks!

Answer: 
Mike Boyle once compiled a pretty large collection of [open problems in symbolic dynamics](http://www.math.umd.edu/%7Emboyle/open/), and has been keeping track of their status.
You mentioned substitution systems, so the Pisot substitution conjecture obviously has to be mentioned. There's a (mostly) up to date exposition by Akiyama, Barge, Berthé, Lee and Siegel that can be read [here](https://www.irif.fr/%7Eberthe/Articles/AkiyamaBargeBertheLeeSiegel.pdf).

In a nutshell, it asks whether it is true that an irreducible Pisot substitution on a finite alphabet always gives rise to a subshift which has pure point dynamical spectrum.

The conjecture is known to be true for all two-letter substitutions, but probably the most interesting case that has been solved is by Barge, where he solved the conjecture for all `$\beta$-substitutions' - <https://arxiv.org/abs/1505.04408>