Abstract We study next-to-leading-power (NLP) threshold corrections in colour-singlet production processes, with particular emphasis on Drell-Yan (DY) and single-Higgs production. We assess the quality of the partonic and hadronic threshold expansions for each process up to NNLO. We determine numerically the NLP leading-logarithmic (LL) resummed contribution in addition to the leading-power next-to-next-to-leading logarithmic (LP NNLL) resummed DY and Higgs cross sections, matched to NNLO. We find that the inclusion of NLP logarithms is numerically more relevant than increasing the precision to N3LL at LP for these processes. We also perform an analytical and numerical comparison of LP NNLL + NLP LL resummation in soft-collinear effective theory and direct QCD, where we achieve excellent analytical and numerical agreement once the NLP LL terms are included in both formalisms. Our results underline the phenomenological importance of understanding the NLP structure of QCD cross sections.