  We analyze the $Z$-boson decay $Z\to \gamma\, X$ into a photon ($\gamma$)
plus a hypothetical light boson ($X$) belonging to a dark or secluded sector.
Due to its feeble interactions with Standard Model fields, this dark boson is
behaving as missing energy in the detector. We consider for $X$ the cases of
spin-1 (massless dark-photon), spin-0 (axion-like), and spin-2 (graviton-like)
particles and explore the way to untangle its spin origin. All these scenarios
predict a universal signature for this decay, characterized by a single
mono-chromatic photon in the $Z$ center of mass, with energy about half of the
$Z$ mass, plus a neutrino-like missing energy associated to the $X$ boson. We
show that if the $Z\to \gamma\, X$ signal should be discovered at $e^+e^-$
colliders, the angular distribution of the mono-chromatic photon in $e^+e^-\to
Z\to \gamma\, X$ can provide a clean probe to discriminate between the $J=1$
and alternative $J=0/2$ spin nature of the $X$ dark boson.
