  Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via
divisorial fans and K\'ovacs-Schwede-Smith characterization of Du Bois
singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of
complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to
be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an
example of a Cohen-Macaulay Du Bois singularity of complexity one which does
not have rational singularities.
