  For a number field $F$ and a prime number $p$, the $\mathbb{Z}_p$-torsion
module of the Galois group of the maximal abelian pro-$p$ extension of $F$
unramified outside $p$ over $F$, denoted as $\mathcal{T}_p(F)$, is an important
subject in abelian $p$-ramification theory. In this paper we study the group
$\mathcal{T}_2(F)=\mathcal{T}_2(m)$ of the quadratic field $F=\mathbb{Q}(\sqrt{
m})$. Firstly, assuming $m>0$, we prove an explicit $4$-rank formula for
$\mathcal{T}_2(-m)$. Furthermore, applying this formula, we obtain the $4$-rank
density of $\mathcal{T}_2$-groups of imaginary quadratic fields. Secondly, for
$l$ an odd prime, we obtain results about the $2$-divisibility of orders of
$\mathcal{T}_2(\pm l)$ and $\mathcal{T}_2(\pm 2l)$. In particular we find that
$\#\mathcal{T}_2(l)\equiv 2\# \mathcal{T}_2(2l)\equiv h_2(-2l)\bmod{16}$ if
$l\equiv 7\bmod{8}$ where $h_2(-2l)$ is the $2$-class number of
$\mathbb{Q}(\sqrt{-2l})$. We then obtain density results for $\mathcal{T}_2(\pm
l)$ and $\mathcal{T}_2(\pm 2l)$. Finally, based on our density results and
numerical data, we propose distribution conjectures about $\mathcal{T}_p(F)$
when $F$ varies over real or imaginary quadratic fields for any prime $p$, and
about $\mathcal{T}_2(\pm l)$ and $\mathcal{T}_2(\pm 2 l)$ when $l$ varies, in
the spirit of Cohen-Lenstra heuristics. Our conjecture in the
$\mathcal{T}_2(l)$ case is closely connected to Shanks-Sime-Washington's
speculation on the distributions of the zeros of $2$-adic $L$-functions and to
the distributions of the fundamental units.
