  We present state-of-the-art predictions for transverse observables relevant
to colour-singlet production at the LHC, in particular the transverse momentum
of the colour singlet in gluon-fusion Higgs production and in neutral Drell-Yan
lepton-pair production, as well as the $\phi^*_\eta$ observable in Drell Yan.
We perform a next-to-next-to-next-to-leading logarithmic (N$^3$LL) resummation
of such observables in momentum space according to the RadISH formalism,
consistently including in our prediction all constant terms of relative order
$\alpha_s^3$ with respect to the Born, thereby achieving N$^3$LL$^\prime$
accuracy. The calculation is fully exclusive with respect to the Born
kinematics, which allows the application of arbitrary fiducial selection cuts
on the decay products of the colour singlet. We supplement our results with a
transverse-recoil prescription, accounting for dominant classes of
subleading-power corrections in a fiducial setup. The resummed predictions are
matched with fixed-order differential spectra at next-to-next-to-leading order
(NNLO) accuracy. A phenomenological comparison is carried out with 13 TeV LHC
data relevant to the Higgs to di-photon channel, as well as to neutral
Drell-Yan lepton-pair production. Overall, the inclusion of ${\cal
O}(\alpha_s^3)$ constant terms, and to a lesser extent of transverse-recoil
effects, proves beneficial for the comparison of theoretical predictions to
data, leaving a residual theoretical uncertainty in the resummation region at
the 2-5% level for Drell-Yan observables, and 5-7% in Higgs production.
