The author considers the polynomial ring \(R=K[x,y]\) over an infinite field \(K\), the subspaces \(R_j\) of degree-\(j\) homogeneous polynomials and the Grassmannians \(\mathrm{Grass}(R_j,d)\) parametrizing the vector spaces \(V \subset R_j\) having dimension \(d\) and then the rational curves \(\phi _V : \mathbb P ^1 \rightarrow \mathbb P ^{d-1}\) of degree \(j\). The restricted tangent bundle \(T_V = \phi _V ^* ( T_{\mathbb P ^{d-1}})\) on \(\mathbb P^1\) has a decomposition \(T_V= \bigoplus _{i=1}^{d-1} \mathcal O (k_i)\). By using the Euler exact sequence on \(\mathbb P ^{d-1}\) and the minimal free resolution of \((V) \subset R\), one can see that knowing the splitting type \(k_1, \ldots , k_{d-1}\) of the restricted tangent bundle \(T_V\) is equivalent to knowing the Betti numbers of \((V) \subset R\) or equivalently to knowing the column degrees of the \(d \times (d-1)\) Hilbert-Burch relation matrix among the generators of \(V\). That is also equivalent to knowing the Hilbert function of the algebra \(R/(V)\). A purpose of this paper is to recall, illustrate and extend some of the properties of the stratification of \(\mathrm{Grass}(R_j,d)\) by the Hilbert functions that the author studied in [Mem. Am. Math. Soc. 188, 112 p. (1977; Zbl 0355.14001)] and [J. Algebra 272, No. 2, 530–580 (2004; Zbl 1119.13015)]. A finer stratification determined by singularities of the rational curves is considered by D. Cox, A. Kustin, C. Polini and B. Ulrich in [Mem. Am. Math. Soc. 1045, v-ix, 116 p. (2013; Zbl 1305.14014)], the methods illustrated by the author would be extend to this stratification and the author poses some open questions. A description of the strata defined by the splitting type of the restricted tangent bundle of rational curves in \(\mathbb P^n\) can be found in [L. Ramella, C. R. Acad. Sci., Paris, Sér. I 311, No. 3, 181–184 (1990; Zbl 0721.14014)]. Furthermore the author generalizes results of D. A. Cox et al. [Comput. Aided Geom. Des. 15, No. 8, 803–827 (1998; Zbl 0908.68174)] and C. D’Andrea [Commun. Algebra 32, No. 1, 159–165 (2004; Zbl 1058.14070)] concerning the dimension and closure of \(\mu\) families of parametrized rational curves from planar to higher dimensional embeddings.