Strategies for the discovery of single-nucleotide polymorphisms (SNPs) can be characterized by the number of individuals in the discovery sample, and by the minimal required number of observations of each allele. We examine the effect of different strategies on two key properties of the resulting SNP collection: (1) the probability that a SNP with a given population allele frequency is detected; and (2) the allele-frequency distribution of the discovered SNPs. We show that strategies that accept all polymorphic sites lead to collections with a high fraction of SNPs with rare minor alleles, particularly in expanded populations. Such SNPs have a low probability of replication in a second sample. We discuss how to tailor a discovery strategy to the desired properties of a SNP collection.