  Neutron stars are identified as pulsars, X-ray binary components, central
objects of supernovae remnants, or isolated thermally emitting sources, and at
distances beyond 120 pc. A population extrapolation suggests 10$^3$ objects
within that boundary. Potentially, neutron stars could continuously emit
gravitational waves at sensitivity reach of present instrumentation. As part of
our Search for the Nearest Neutron Stars ``Five Seasons'' project, we search
for nearby resolved neutron stars. Based on expected fluxes and magnitudes of
thermally cooling neutron stars and pulsars, we selected sources in Gaia DR3.
The sources have $G$-band absolute magnitudes $M_G>16$ mag, parallax
signal-to-noise ratios greater than two, and colours $G_{BP}-G<0.78$ and
$G-G_{RP}<0.91$ mag for power-law emitters of flux $F_{\nu} \propto
\nu^{-\alpha_{\nu}}$ with spectral indices $\alpha_{\nu}<3$. The photometric
region overlaps with that of white dwarfs, in confluence with most known
pulsars in binaries having white dwarf companions. We looked for counterparts
in gamma-ray, X-ray, ultraviolet, radio, optical, and infrared catalogues. We
find about two X-ray-, 15 ultraviolet-, one radio probable counterparts, and at
least four sources with power-law profiles at the
ultraviolet-optical(-infrared). Because the sources have $G\gtrapprox20$ mag,
we rely on Gaia DR3 single-source parameters. We identify possible binaries
based on photoastrometric parameters, visual companions, and flux excesses.
Some emission components suggest small thermal radii. Source types, neutron
star content, and properties require further inquiry.
