**Update.**

Estimates coming from the Bogomolov-Miyaoka-Yau inequality yield:

**Corollary.** (to Theorem A in this paper of Sakai) There is no $M$ such that every curve admits an injective morphism to a curve in $\mathbb{P}^2$ whose points all have multiplicity at most $M$.

*Proof.* The geometric genus $g$ curves of degree $d$ move in a family of dimension $3d + g - 1$, so if these are to fill the moduli space we must have $2g \le 3d + 2$. On the other hand, Theorem A of this paper of Sakai gives after rearranging that for any genus $g$, degree $d$, unibranch plane curve,
$$ 2g \ge \frac{d^2}{2m+1} + O(md) $$
where $m$ is the maximal multiplicity of a point on the curve. For any bounded $m$, these inequalities cannot both be satisfied for large $g$. $\square$

Remark: the argument also shows that the same statement holds restricted to hyperelliptic curves. This is not incompatible with Jeremy's answer below, which requires a cusp of multiplicity $2g$ to inject a hyperelliptic curve of genus $g$.

I leave the older argument below because it is a different and possibly still useful approach.

**Theorem.** There is no finite set $\Sigma$ of topological types of unibranch singularities such that every smooth curve admits an injective map to a curve in $\mathbb{P}^2$ with only singularities from $\Sigma$.

*Proof.* I will argue that curves with unibranch singularities of total cogenus $\delta$ occur in the linear system $|\mathcal{O}(d)|$ with codimension at least $\mathrm{min}(\kappa, 3d - 1 + \delta)$, where $\kappa$ is the codimension of the equiclassical locus in the versal deformation of the singularities; it is $2\delta$ in the case of curves with only cusps, and for any finite set of singularities is bounded below by $(1 + \epsilon)\delta$ for $0 < \epsilon \le 1$ some constant depending on the set of singularities. Let us first see why this implies the result.

Saying a geometric genus $g$ curve appears in degree $d$ in codimension $3d - 1 + \delta = 3d + (d-1)(d-2)/2 - g - 1$ means it comes in a $g + 1 + (d+1)(d+2)/2 - (d-1)(d-2)/2 -3d = g + 3$ dimensional family. On the other hand $\kappa$ is given by $\delta + r$, where $r$ is the total ramification divisor on the normalization of the curve (for the normalization map), i.e. the sum over the branches of the singularities of (multiplicity - 1). So in this case the curves come in a family of dimension $g + 4 + 3d - r$. *I don't know what to do with this,* but under the further assumption that the set of singularities from which we choose must be finite and so we get $\kappa = (1 + \epsilon) \delta$ as above (e.g., for only cusps, $\epsilon = 1$), then the argument in Will's answer above shows this can fill the moduli space for only finitely many $g$.

To get the advertised codimension estimate, let $C$ be such a curve of degree $d$. Let $\Lambda$ be the tangent space to the versal deformation of the singularities of $C$. Then there exists a linear subspace $L \subset \Lambda$ such that

- equigeneric ideal $\ge$ L $\ge$ equiclassical ideal
- aside from the above restriction, L may be taken so that the dimension of the equigeneric ideal mod L is up to $3d - 1$.
- the image of $T_C|\mathcal{O}(d)| \to \Lambda$ is transverse to $L$.

This is essentially classical, for the sort of argument to prove it you can see e.g. the statement of Lemma 8 and the proof of Lemma 10 in my paper with Steve. $\square$

It's not yet clear to me how to improve this to answer the OP, maybe the better sort of bounds of Greuel, Lossen, and Shustin will help? (Or maybe there's just some easy argument not involving deformation theory...)