  A singular point on a plane conic defined over $\mathbb{Q}$ is a
transcendental point of the curve which admits very good rational
approximations, uniformly in terms of the height. Extremal numbers and Sturmian
continued fractions are abscissa of such points on the parabola $y=x^2$. In
this paper we provide a measure of transcendence for singular points on conics
defined over $\mathbb{Q}$ which, in these two cases, improves on the measure
obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of
Schmidt subspace theorem due to Evertse.
