We develop statistically based methods to detect single nucleotide DNA mutations in next generation sequencing data. Sequencing generates counts of the number of times each base was observed at hundreds of thousands to billions of genome positions in each sample. Using these counts to detect mutations is challenging because mutations may have very low prevalence and sequencing error rates vary dramatically by genome position. The discreteness of sequencing data also creates a difficult multiple testing problem: current false discovery rate methods are designed for continuous data, and work poorly, if at all, on discrete data. We show that a simple randomization technique lets us use continuous false discovery rate methods on discrete data. Our approach is a useful way to estimate false discovery rates for any collection of discrete test statistics, and is hence not limited to sequencing data. We then use an empirical Bayes model to capture different sources of variation in sequencing error rates. The resulting method outperforms existing detection approaches on example data sets.