  Motivated by recent applications in rough volatility and regularity
structures, notably the notion of singular modelled distribution, we study
paths, rough paths and related objects with a quantified singularity at zero.
In a pure path setting this allows us to leverage on existing SLE Besov
estimates to see that SLE traces takes values in a singular H\"older space,
which quantifies a well-known boundary effect in the regime $\kappa < 1$. We
then consider the integration theory against singular rough paths and some
extensions thereof. This gives a method to reconcile, from a regularity
structure point of view, different singular kernels used to construct
(fractional) rough volatility models and an effective reduction to the
stationary case which is crucial to apply general renormalisation methods.
