  This work addresses the {\em singularity formation} of complete non-compact
solutions to the conformally flat Yamabe flow whose conformal factors have {\em
cylindrical behavior at infinity}. Their singularity profiles happen to be {\em
Yamabe solitons}, which are {\em self-similar solutions} to the fast diffusion
equation satisfied by the conformal factor of the evolving metric. The
self-similar profile is determined by the second order asymptotics at infinity
of the initial data which is matched with that of the corresponding
self-similar solution. Solutions may become extinct at the extinction time $T$
of the cylindrical tail or may live longer than $T$. In the first case the
singularity profile is described by a {\em Yamabe shrinker} that becomes
extinct at time $T$. In the second case, the singularity profile is described
by a {\em singular} Yamabe shrinker slightly before $T$ and by a matching {\em
Yamabe expander} slightly after $T$ .
