This paper extends False Discovery Rates to random fields, where there are uncountably many hypothesis tests. This provides a method for finding local regions in the field where there is a significant signal while controlling either the proportion of area or the number of clusters in which false rejections occur. We develop confidence envelopes for the proportion of false discoveries as a function of the rejection threshold. This yields algorithms for constructing a confidence superset for the locations of the true nulls. From this we derive rejection thresholds that control the mean and quantiles of the proportion of false discoveries. We apply this method to scan statistics and functional neuroimaging.