We study composition operators on Hardy and Dirichlet spaces belonging to Schatten classes. We give some new examples and analyse the size of contact set of the symbol of such operators.The contact set of ϕ is E ϕ := E ϕ (1). Let H be a Hillbert space, a compact operator is said to belong in the Schatten class S p (H ) if its sequence of singular numbers is in the sequence space ℓ p .2000 Mathematics Subject Classification. 47B38, 30H05, 30C85, 47A15.