  We carry out extensive tests of the next-to-leading logarithmic (NLL)
accuracy of the PanScales parton showers, as introduced recently for
colour-singlet production in hadron collisions. The tests include comparisons
to (semi-)analytic NLL calculations of a wide range of hadron-collider
observables: the colour-singlet boson transverse momentum distribution; global
and non-global hadronic energy flow variables related to jet vetoes and
analogues of jettiness distributions; (sub)jet multiplicities; and observables
sensitive to the DGLAP evolution of the incoming momentum fractions. In the
tests, we also include an implementation of a standard transverse-momentum
ordered dipole shower, to establish the size of missing NLL effects in such
showers, which, depending on the observable, can reach $100\%$. This paper,
together with arXiv:2205.02237, constitutes the first step towards
process-independent NLL-accurate parton showers for hadronic collisions.
