We propose local segmentation of multiple sequences sharing a common time- or location-index, building upon the single sequence local segmentation methods of Niu and Zhang (2012) and Fang, Li and Siegmund (2016). We also propose reverse segmentation of multiple sequences that is new even in the single sequence context. We show that local segmentation estimates change-points consistently for both single and multiple sequences, and that both methods proposed here detect signals well, with the reverse segmentation method outperforming a large number of known segmentation methods on a commonly used single sequence test scenario. We show that on a recent allele-specific copy number study involving multiple cancer patients, the simultaneous segmentations of the DNA sequences of all the patients provide information beyond that obtained by segmentation of the sequences one at a time.