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 Córdoba-Córdoba-Fontelos (CCF) model.