  We discuss with a rather critical eye the current situation of black hole
(BH) solutions in $f(R)$ gravity and shed light about its geometrical and
physical significance. We also argue about the meaning, existence or lack
thereof of a Birkhoff's theorem in this kind of modified gravity. We focus then
on the analysis and quest of $non-trivial$ (i.e. hairy)
$asymptotically\,\,flat$ (AF) BH solutions in static and spherically symmetric
(SSS) spacetimes in vacuum having the property that the Ricci scalar does $not$
vanish identically in the domain of outer communication. To do so, we provide
and enforce the $regularity\,\,conditions$ at the horizon in order to prevent
the presence of singular solutions there. Specifically, we consider several
classes of $f(R)$ models like those proposed recently for explaining the
accelerated expansion in the universe and which have been thoroughly tested in
several physical scenarios. Finally, we report analytical and numerical
evidence about the $absence$ of $geometric\,\,hair$ in AFSSSBH solutions in
those $f(R)$ models. First, we submit the models to the available no-hair
theorems, and in the cases where the theorems apply, the absence of hair is
demonstrated analytically. In the cases where the theorems do not apply, we
resort to a numerical analysis due to the complexity of the non-linear
differential equations. Within that aim, a code to solve the equations
numerically was built and tested using well know exact solutions. In a future
investigation we plan to analyze the problem of hair in De Sitter and Anti-De
Sitter backgrounds.
