  The kagome Heisenberg antiferromagnet formed by frustrated spins arranged in
a lattice of corner-sharing triangles is a prime candidate for hosting a
quantum spin liquid (QSL) ground state consisting of entangled spin singlets.
But the existence of various competing states makes a convincing theoretical
prediction of the QSL ground state difficult, calling for experimental clues
from model materials. The kagome lattice materials Zn-barlowite
ZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do not
exhibit long range order, and they are considered the best realizations of the
kagome Heisenberg antiferromagnet known to date. Here we use $^{63}$Cu nuclear
quadrupole resonance combined with the inverse Laplace transform (ILT) to probe
locally the inhomogeneity of delicate quantum ground states affected by
disorder. We present direct evidence for the gradual emergence of spin singlets
with spatially varying excitation gaps, but even at temperatures far below the
super-exchange energy scale their fraction is limited to approximately 60\% of
the total spins. Theoretical models need to incorporate the role of disorder to
account for the observed inhomogeneously gapped behaviour.
