  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
N$^3$LL 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.
