  In this paper, we establish a local theory, i.e., existence, uniqueness and
blow-up criterion, for a general family of singular SDEs in some Hilbert space.
The key requirement is an approximation property that allows us to embed the
singular drift and diffusion mappings into a hierarchy of regular mappings that
are invariant with respect to the Hilbert space and enjoy a cancellation
property.
  Various nonlinear models in fluid dynamics with transport noise belong to
this type of singular SDEs. With a cancellation estimate for generalized Lie
derivative operators, we can construct such regular approximations for cases
involving the Lie derivative operators, or more generally, differential
operators of order one with suitable coefficients. In particular, we apply the
abstract theory to achieve novel local-in-time results for the stochastic
two-component Camassa--Holm (CH) system and for the stochastic
C\'ordoba-C\'ordoba-Fontelos (CCF) model.
