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We present state-of-the-art predictions for transverse observables relevantto colour-singlet production at the LHC, in particular the transverse momentumof the colour singlet in gluon-fusion Higgs production and in neutral Drell-Yanlepton-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) resummationof 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 Bornkinematics, which allows the application of arbitrary fiducial selection cutson the decay products of the colour singlet. We supplement our results with atransverse-recoil prescription, accounting for dominant classes ofsubleading-power corrections in a fiducial setup. The resummed predictions arematched with fixed-order differential spectra at next-to-next-to-leading order(NNLO) accuracy. A phenomenological comparison is carried out with 13 TeV LHCdata relevant to the Higgs to di-photon channel, as well as to neutralDrell-Yan lepton-pair production. Overall, the inclusion of ${\calO}(\alpha_s^3)$ constant terms, and to a lesser extent of transverse-recoileffects, proves beneficial for the comparison of theoretical predictions todata, leaving a residual theoretical uncertainty in the resummation region atthe 2-5% level for Drell-Yan observables, and 5-7% in Higgs production.

Fiducial Higgs and Drell-Yan distributions at N$^3$LL$^\prime$+NNLO with RadISH