Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?
Magma has CanonicalMap and CanonicalImage, which can be used to compute the canonical model (which is smooth if the curve is not hyperelliptic). This should work for reasonable genus and size of coefficients. You can use IsHyperelliptic and IsGeometricallyHyperelliptic to get a hyperelliptic model (smooth in a weighted projective plane) when the curve is hyperelliptic.