  The aim of this paper is to prove some classification results for generic
shrinking Ricci solitons. In particular, we show that every three dimensional
generic shrinking Ricci soliton is given by quotients of either $\mathds{S}^3$,
$\erre\times\mathds{S}^2$ or $\erre^3$, under some very weak conditions on the
vector field $X$ generating the soliton structure. In doing so we introduce
analytical tools that could be useful in other settings; for instance we prove
that the Omori-Yau maximum principle holds for the $X$-Laplacian on every
generic Ricci soliton, without any assumption on $X$.
