CCGs are directly compatible with binary-branching bottom-up parsing algorithms, in particular CKY and shift-reduce algorithms. While the chart-based approach has been the dominant approach for CCG, the shift-reduce method has been little explored. In this paper, we develop a shift-reduce CCG parser using a discriminative model and beam search, and compare its strengths and weaknesses with the chart-based C&C parser. We study different errors made by the two parsers, and show that the shift-reduce parser gives competitive accuracies compared to C&C. Considering our use of a small beam, and given the high ambiguity levels in an automatically-extracted grammar and the amount of information in the CCG lexical categories which form the shift actions, this is a surprising result.