  We have studied the noncommutative extension of the relativistic
Chern-Simons-Higgs model, in the first non-trivial order in $\theta$, with only
spatial noncommutativity. Both Lagrangian and Hamiltonian formulations of the
problem have been discussed, with the focus being on the canonical and
symmetric forms of the energy-momentum tensor. In the Hamiltonian scheme,
constraint analysis and the induced Dirac brackets have been provided. The
spacetime translation generators and their actions on the fields are discussed
in detail.
  The effects of noncommutativity on the soliton solutions have been analysed
thoroughly and we have come up with some interesting observations. Considering
the {\it{relative}} strength of the noncommutative effects, we have shown that
there is a universal character in the noncommutative correction to the magnetic
field - it depends {\it{only}} on $\theta$. On the other hand, in the cases of
all other observables of physical interest, such as the potential profile,
soliton mass or the electric field, $\theta$ as well as $\tau$, (comprising
solely of commutative Chern-Simons-Higgs model parameters), appear on similar
footings. This phenomenon is a new finding which has come up in the present
analysis.
  Lastly, we have pointed out a generic problem in the NC extension of the
models, in the form of a mismatch between the BPS dynamical equation and the
full variational equations of motion, to $O(\theta)$. This mismatch indicates
that the analysis is not complete as it brings in to fore the ambiguities in
the definition of the energy-momentum tensor in a noncommutative theory.
