Criteria are found which classify Schatten class membership for operators on the Hardy space in terms of their action on the reproducing kernels and their derivatives. The Hilbert–Schmidt conditions are shown to be both necessary and sufficient for arbitrary operators. For other Schatten classes, the criteria are shown to be either necessary or sufficient for arbitrary operators, depending upon the exponent of the class. However, using known results of Peller and Luecking, the criteria are shown to be both necessary and sufficient for Hankel operators and Carleson embeddings.