  In this paper we study the behaviour of selective separability properties in
the class of Frech\'{e}t-Urysohn spaces. We present two examples, the first one
given in ZFC proves the existence of a countable Frech\'{e}t-Urysohn (hence
$R$-separable and selectively separable) space which is not $H$-separable;
assuming $\mathfrak{p}=\mathfrak{c}$, we construct such an example which is
also zero-dimensional and $\alpha_{4}$. Also, motivated by a result of Barman
and Dow stating that the product of two countable Frech\'{e}t-Urysohn spaces is
$M$-separable under PFA, we show that the MA is not sufficient here. In the
last section we prove that in the Laver model, the product of any two
$H$-separable spaces is $mH$-separable.
