We deal with some cardinality issues concerning the set of critical angles of a closed and convex cone in the Euclidean space. Critical angles are stationary points associated to the optimization problem consisting of finding a pair of points belonging to the cone, with a maximal angle between them. This set is called the angular spectrum of the cone. For a polyhedral cone, we have proved that the angluar spectrum is finite, and its cardinality is bounded by a polynomial in the number of generators. In this paper we construct (non-polyhedral) cones with both countable and noncountable angular spectra.