  Gravity modes (g modes), mixed gravito-acoustic modes (mixed modes), and
gravito-inertial modes (gi modes) possess unmatched properties as probes for
stars with radiative interiors. The structural and dynamical constraints that
they are able to provide cannot be accessed by other means. While they provide
precious insights into the internal dynamics of evolved stars as well as
massive and intermediate-mass stars, their non-detection in main sequence (MS)
solar-type stars make them a crucial missing piece in our understanding of
angular momentum transport in radiative zones and stellar rotational evolution.
In this work, we aim to apply certain analysis tools originally developed for
helioseismology in order to look for g-mode signatures in MS solar-type stars.
We select a sample of the 34 most promising MS solar-type stars with Kepler
four-year long photometric time series. All these stars are well-characterised
late F-type stars with thin convective envelopes, fast convective flows, and
stochastically excited acoustic modes (p modes). For each star, we compute the
background noise level of the Fourier power spectrum to identify significant
peaks at low frequency. After successfully detecting individual peaks in 12
targets, we further analyse four of them and observe distinct patterns of
surrounding peaks with a low probability of being noise artifacts. Comparisons
with the predictions from reference models suggest that these patterns are
compatible with the presence of non-asymptotic low-order pure g modes, pure p
modes, and mixed modes. Given their sensitivity to both the convective core
interface stratification and the coupling between p- and g-mode resonant
cavities, such modes are able to provide strong constraints on the structure
and evolutionary states of the related targets. [abridged]
