Given a smooth curve defined over a field k that admits a non-singular plane model over k, a fixed separable closure of k, it does not necessarily have a non-singular plane model defined over the field k. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over k for its twists. We characterize twists possessing such models and use such characterization to improve, for the particular case of smooth plane curves, the algorithm to compute twists of non-hyperelliptic curves wrote recently down by the third author. We also show an example of a twist not admitting such non-singular plane model. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to compute all the twists of smooth plane curves with cyclic automorphism group having a k-model whose automorphism group is generated by a diagonal matrix. Some examples are also provided.