  We study the linear stability of Einstein metrics of Riemannian submersion
type. First, we derive a general instability condition for such Einstein
metrics and provide some applications. Then we study instability arising from
Riemannian product structures on the base. As an application, we estimate the
coindex of the Einstein metrics constructed in \cite{WZ90} and \cite{Wan92}.
Finally, we investigate more closely the linear stability of Einstein metrics
from circle bundle constructions and obtain a rigidity result for linearly
stable Einstein metrics of this type.
