  This paper serves as a first foray on regularisation for planar vector
fields. Motivated by singularities in celestial mechanics, the block
regularisation of a generic class of degenerate singularities is studied. The
paper is concerned with asymptotic properties of the transition map between a
section before and after the singularity. Block regularisation is reviewed
before topological and explicit conditions for the $ C^0 $-regularity of the
map are given. Computation of the $ C^1 $-regularisation is reduced to summing
residues of a rational function. It is shown that the transition map is in
general only finitely differentiable and a method of computing the map is
conveyed. In particular, a perturbation of a toy example derived from the
4-body problem is shown to be $ C^{4/3} $. The regularisation of all
homogeneous quadratic vector fields is computed.
