We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type $A$ and type $B$. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.
Keywords: Dyck paths, cyclic sieving, Narayana numbers, major index, q-analog.
Mathematics Subject Classifications: 05E18, 05A19, 05A30