In this talk I will address the dichotomy between amenability and the existence of paradoxical decompositions for inverse semigroups. Semigroups have singular dynamics in the sense that multiplication by an element can drastically shrink the size of a finite subset in the semigroup. This fact forces to analyze again basic aspects coming from group theory like paradoxical decompositions, Foelner nets or existence of invariant means. We will analyze these aspects in relation to two C*-algebras that generalize the usual reduced group C*-algebra as well as the uniform Roe algebra of a countable finitely generated group.