  Nuclear spins are among the potential candidates prospected for quantum
information technology. A recent breakthrough enabled to atomically resolve
their interaction with the electron spin, the so-called hyperfine interaction,
within individual atoms utilizing scanning tunneling microscopy (STM).
Intriguingly, this was only realized for a few species put on a two-layers
thick MgO. Here, we systematically quantify from first-principles the hyperfine
interactions of the whole series of 3d transition adatoms deposited on various
thicknesses of MgO, NaF, NaCl, h--BN and Cu$_2$N films. We identify the
adatom-substrate complexes with the largest hyperfine interactions and unveil
the main trends and exceptions. We reveal the core mechanisms at play, such as
the interplay of the local bonding geometry and the chemical nature of the thin
films, which trigger transitions between high- and low-spin states accompanied
with subtle internal rearrangements of the magnetic electrons. By providing a
general map of hyperfine interactions, our work has immediate implications in
future STM investigations aiming at detecting and realizing quantum concepts
hinging on nuclear spins.
