  We reconstruct the rank-one, singular (point-like) perturbations of the
$d$-dimensional fractional Laplacian in the physically meaningful
norm-resolvent limit of fractional Schr\"{o}dinger operators with regular
potentials centred around the perturbation point and shrinking to a delta-like
shape. We analyse both the possible regimes, the resonance-driven and the
resonance-independent limit, depending on the power of the fractional Laplacian
and the spatial dimension. To this aim, we also qualify the notion of
zero-energy resonance for Schr\"{o}dinger operators formed by a fractional
Laplacian and a regular potential.
