We develop a variational method of deriving stochastic partial differential equations whose solutions follow
the flow of a stochastic vector field. As an example in one spatial dimension we numerically simulate singular
solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method.
These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation
persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity
to finite dimensional solutions of stochastic partial differential equations (SPDE). In particular, some choices of
stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations,
or CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although
this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincar´e
structure of the CH equation (parametric stochastic deformations, or P-SD), and it also does not occur for
peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science
of stochastic deformations of finite-dimensional approximations of evolutionary PDE and the sensitivity of the
resulting solutions to the choices made in stochastic modelling.