Threshold dose/concentration values, such as the lowest effective dose, minimum effective dose or the lowest effective concentration (LED, MED or LEC, respectively) are in use as an alternative to the mutagen potency measures based on the 'rate' measurements (e.g., the slope of the initial part of the dose-response curve). In this respect, several statistical procedures for the corresponding so-called 'dose finding' were proposed during the last decades. However, most of them disregard the discrete nature of responses such as the plate colony count in the Ames Salmonella assay. When the plate counts agree with the Poisson assumption, two procedures considered here seem to be appropriate for the dose finding. One is based on the stepwise collapsing of the homogeneous control and dose counts; another consists of constructing the confidence limits for the mutation induction factor (MIF). When the dose and control counts are non-overlapping, the simple 'visual' non-parametric estimation of LED is possible. Applicability and validity of the methods is demonstrated with the two data sets on the mutagenicity of the beta-carboline alkaloid, harmine, and one of the oxidation products of apomorphine.