  The present paper deals with the investigation of the structure of general
fiber product rings $R\times_TS$, where $R$, $S$ and $T$ are local rings with
common residue field. We show that the Poincar\'e series of any $R$-module over
the fiber product ring $R\times_TS$ is bounded by a rational function. In
addition, we give a description of ${\rm depth}(R\times_TS)$, which is an open
problem in this theory. As a biproduct, using the characterization of the Betti
numbers over $R\times_TS$ obtained, we provide certain cases of the
Cohen-Macaulayness of $R\times_TS$ and, in particular, we show that
$R\times_TS$ is always non-regular. Some positive answers for the
Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over $R\times_TS$ are
also established.
