Prevalent singular cardinals hypothesis The following notion is introduced by Assaf Rinot:
Definition. A singular cardinal $\kappa$  is a prevalent singular
cardinal iff there exists a family $\mathbb{A}\subset  P(\kappa)$ with $|\mathbb{A}| = \kappa$
and $sup\{|A| : A\in \mathbb{A}\} < \kappa$ such that any $B\subset \kappa$   with
$|B| < cf(\kappa)$ is contained in some $A\in \mathbb{A}$.
Prevalent Singular Cardinals Hypothesis (PSCH) states that any singular cardinal is a prevalent singular cardinal.
Clearly any singular cardinal of countable cofinality is prevalent.
Question. What is known about the consistency of the failure of PSCH?
Remark 1. PSCH follows from $GCH, SCH, PFA$ and many other combinatorial principles. 
Remark 2. As stated in Assaf Rinot's paper "On topological spaces of singular density and minimal weight, Topology and its Applications 155 (2007) 135–140", $PSCH$ is a very weak assertion and all currently known methods for violating statements
of similar flavor, will fail to violate the $PSCH$.
Question 2. Is there any relation between PSCH and some combinatorial principles introduced by Shelah in PCF theory like $Cov$ or ...?
 A: A singular cardinal $\kappa$ is prevalent iff
$\operatorname{cov} (\kappa , \mu , \operatorname{cf} (\kappa)) = \kappa$
for some cardinal $\mu$ with
$\operatorname{cf} (\kappa) \leq \mu < \kappa$.
Proof:
First, it is easy to show that
$\operatorname{cov} (\kappa , \mu , \theta) \geq \kappa$
when $\theta \leq \mu < \kappa$.
Now, if $\kappa$ is prevalent, then
$\operatorname{cov} (\kappa , \mu , \operatorname{cf} (\kappa)) \leq \kappa$,
with $\eta = \operatorname{sup} \{ |A| : A \in \mathbb{A} \}$ and
$$
\mu =
\left\lbrace
\begin{array}{ll}
\eta , & \textrm{if } |A| < \eta \textrm{ for all } A \in \mathbb{A} ; \\
\eta^{+} , & \textrm{otherwise.}
\end{array}
\right.
$$
The other direction is immediate. $\square$
SSH (Shelah's Strong Hypothesis) implies SCH, hence implies PSCH.
By Corollary 3.6 in

Pierre Matet. Large cardinals and covering numbers,
  Fundamenta Mathematicae 205 (2009), 45-75.
  doi:10.4064/fm205-1-3

SSH is equivalent to the following: given cardinals $\mu$ and $\lambda$,
with $\mu \geq \lambda = \operatorname{cf} (\lambda) \geq \aleph_{1}$,
$$
\operatorname{cov}  \left( \mu , \lambda , \lambda \right) =
\left\lbrace
\begin{array}{ll}
\mu , & \textrm{if } \operatorname{cf} (\mu) \geq \lambda ; \\
\mu^{+} , & \textrm{otherwise.}
\end{array}
\right.
$$
Under SSH, when $\kappa$ is a singular cardinal with
$\operatorname{cf} (\kappa) > \aleph_0$, we have, for any $\mu$ with
$\operatorname{cf} (\kappa) \leq \mu < \kappa$:
$$
\operatorname{cov} (\kappa , \mu , \operatorname{cf} (\kappa)) \leq
\operatorname{cov} (\kappa , \operatorname{cf} (\kappa) , \operatorname{cf} (\kappa)) = \kappa .
$$
Note that the condition (weaker than prevalent)
$\operatorname{cov} (\kappa , \kappa , \operatorname{cf} (\kappa)) \leq \kappa$
(for a singular cardinal $\kappa$)
is a theorem in ZFC. In fact,
$\operatorname{cov} (\kappa , \kappa , \operatorname{cf} (\kappa)) = \operatorname{cf} (\kappa)$,
by Observation 5.2(3) in Chapter II of

S. Shelah. Cardinal Arithmetic,
  volume 29 of Oxford Logic Guides. Oxford University Press, New York, 1994.

See Definition 5.1 in Chapter II for
$\operatorname{cov} (\lambda , \kappa , \theta , \sigma)$.
Then,
$\operatorname{cov} (\lambda , \kappa , \theta) :=
\operatorname{cov} (\lambda , \kappa , \theta , 2)$.
Finally, the condition "$|B| < \operatorname{cf} (\kappa)$"
in the definition of "prevalent singular cardinal"
cannot be replaced by "$|B| \leq \operatorname{cf} (\kappa)$",
because, for any singular cardinal $\kappa$ and any $\mu$ with
$\operatorname{cf} (\kappa) < \mu < \kappa$ we have
$$
\operatorname{cov} (\kappa , \mu , {(\operatorname{cf} (\kappa))}^+) \geq
\operatorname{cov} (\kappa , \kappa , {(\operatorname{cf} (\kappa))}^+) \geq
\operatorname{cov} (\kappa , \kappa , {(\operatorname{cf} (\kappa))}^+ , \operatorname{cf} (\kappa)) \geq {\kappa}^+ ,
$$
by Fact 1 in

Andreas Liu, Bounds for covering numbers,
  The Journal of Symbolic Logic 71 (2006), 1303-1310.
  doi:10.2178/jsl/1164060456

A: A singular cardinal $\lambda$ is prevalent iff there exists some cardinal $\mu<\lambda$ such that $Cov(\lambda,\mu,cf(\lambda),2)=\lambda$. In his solution of the pcf conjecture, Gitik has constructed a model where there exists a singular cardinal $\lambda$ of cofinality $\aleph_1$ such that $pp(\mu)>\lambda$ for cofinally many $\mu<\lambda$ of countable cofinality. Given $\mu<\lambda$, pass to a bigger $\mu'<\lambda$ of countable cofinality with $pp(\mu')>\lambda$. Then, we have $pp(\mu')\le Cov(\mu',\mu',cf(\mu')^+,2)\le Cov(\lambda,\mu',cf(\lambda),2)\le Cov(\lambda,\mu,cf(\lambda),2)$. So $Cov(\lambda,\mu,cf(\lambda),2)>\lambda$ for every $\mu<\lambda$, meaning that $\lambda$ is not prevalent.
