  We perform hydrodynamic simulations using the method of multi-particle
collision dynamics and a theoretical analysis to study a single squirmer
microswimmer at high P\'eclet number, which moves in a low Reynolds number
fluid and under gravity. The relevant parameters are the ratio $\alpha$ of
swimming to bulk sedimentation velocity and the squirmer type $\beta$. The
combination of self-propulsion, gravitational force, hydrodynamic interactions
with the wall, and thermal noise leads to a surprisingly diverse behavior. At
$\alpha > 1$ we observe cruising states, while for $\alpha<1$ the squirmer
resides close to the bottom wall with the motional state determined by stable
fixed points in height and orientation. They strongly depend on the squirmer
type $\beta$. While neutral squirmers permanently float above the wall with
upright orientation, pullers float for $\alpha$ larger than a threshold value
$\alpha_{\mathrm{th}}$ and are pinned to the wall below $\alpha_{\mathrm{th}}$.
In contrast, pushers slide along the wall at lower heights, from which thermal
orientational fluctuations drive them into a recurrent floating state with
upright orientation, where they remain on the timescale of orientational
persistence.
