  Weakly stable torsion classes were introduced by the author and Yekutieli to
provide a torsion theoretic characterisation of the notion of weak
proregularity from commutative algebra. In this paper we investigate weakly
stable torsion classes, with a focus on aspects related to localisation and
completion. We characterise when torsion classes arising from left denominator
sets and idempotent ideals are weakly stable. We show that every weakly stable
torsion class $\operatorname{\mathsf{T}}$ can be associated with a dg ring
$A_{\operatorname{\mathsf{T}}}$; in well behaved situations there is a
homological epimorphism $A\to A_{\operatorname{\mathsf{T}}}$. We end by
studying torsion and completion with respect to a single regular and normal
element.
