In this talk, I will focus on the class of cluster algebras of finite type, which are classified by Dynkin diagrams. We study these commutative algebras from the point of view of singularity theory: we classify their singularities and develop constructive resolutions of these singularities over fields of arbitrary characteristics. From the same perspective, we study cluster algebras coming from a star shaped quiver, which are not of finite type, but whose singularities exhibit interesting combinatorial phenomena. This is joint work with Angélica Benito, Hussein Mourtada, and Bernd Schober.