  We consider Drell-Yan production $pp\to V^* X \to L X$ at small $q_T \ll Q$.
Experimental measurements require fiducial cuts on the leptonic state $L$,
which introduce enhanced, linear power corrections in $q_T/Q$. We show that
they can be unambiguously predicted from factorization, and resummed to the
same order as the leading-power contribution. We thus obtain predictions for
the fiducial $q_T$ spectrum to N3LL and next-to-leading-power in $q_T/Q$.
Matching to full NNLO ($\alpha_s^2$), we find that the linear power corrections
are indeed the dominant ones, and the remaining fixed-order corrections become
almost negligible below $q_T \lesssim 40$ GeV. We also discuss the implications
for more complicated observables, and provide predictions for the fiducial
$\phi^*$ spectrum at N3LL+NNLO. We find excellent agreement with ATLAS and CMS
measurements of $q_T$ and $\phi^*$. We also consider the $p_T^\ell$ spectrum.
We show that it develops leptonic power corrections in $q_T/(Q - 2p_T^\ell)$,
which diverge near the Jacobian peak $p_T^\ell \sim Q/2$ and must be kept to
all powers to obtain a meaningful result there. Doing so, we obtain for the
first time an analytically resummed result for the $p_T^\ell$ spectrum around
the Jacobian peak at N3LL+NNLO. Our method is based on performing a complete
tensor decomposition for hadronic and leptonic tensors. In practice this is
equivalent to often-used recoil prescriptions, for which our results now
provide rigorous, formal justification. Our tensor decomposition yields nine
Lorentz-scalar hadronic structure functions, which directly map onto the
commonly used angular coefficients, but also holds for arbitrary leptonic final
states. In particular, for suitably defined Born-projected leptons it still
yields a LO-like angular decomposition even when including QED final-state
radiation. We also discuss the application to $q_T$ subtractions.
