In this paper we investigate numerous constructions of minimal systems from the point of view of $(\mathcal {F}_1,\mathcal {F}_2)$-chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$). We consider standard es of systems, such as Toeplitz flows, Grillenberger $K$-systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results answer a few open problems known from the literature.