Sequencing studies of breast tumour cohorts have identified many prevalent mutations, but provide limited insight into the genomic diversity within tumours. Here we developed a whole-genome and exome single cell sequencing approach called nuc-seq that uses G2/M nuclei to achieve 91% mean coverage breadth. We applied this method to sequence single normal and tumour nuclei from an oestrogen-receptor-positive(ER+) breast cancer and a triple-negative ductal carcinoma. In parallel, we performed single nuclei copy number profiling. Our data show that aneuploid rearrangements occurred early in tumour evolution and remained highly stable as the tumour masses clonally expanded. In contrast, point mutations evolved gradually, generating extensive clonal diversity. Using targeted single-molecule sequencing, many of the diverse mutations were shown to occur at low frequencies (<10%) in the tumour mass. Using mathematical modelling we found that the triple-negative tumour cells had an increased mutation rate (13.3x), whereas the ER+ tumour cells did not. These findings have important implications for the diagnosis, therapeutic treatment and evolution of chemoresistance in breast cancer.