In Geometry of Schemes by Eisenbud and Harris, it is claimed in Exercise II-19 that:
There are infinitely many isomorphism types of degree 7 subschemes supported at the origin in 3-space, and infinitely many types of degree 8 subschemes supported at the origin in the plane.
However, this exercise is marked "For those with some familiarity with curves," and I do not seem to have the necessary familiarity, because no ideas come to mind.
How could one use curve theory to approach this problem? We work over $\mathbb C$.
I will describe one of the standard arguments, using linear algebra and let us do this for 3-space. Consider the ring $R=R_0\oplus R_1\oplus R_2$, where $R_i$ is the vector space of forms of degree $i$. This is ten dimensional and any subspace $V\subset R_2$ of dimension 3 gives, by quotienting, a ring of dimension 7 as you seek. Such vector spaces form a 9 dimensional family ($\dim Gr(3,6)$) and if two of these quotients are isomorphic, one can show that there is an automorphism of $R_1$ which induces this. (This requires a small argument). But the dimension of these automorphisms is 9 ($Gl(3)$) and the scalar matrices acts trivially. So, we have at least one dimensional family of non-isomorphic such rings.