= Solution
Write the irreducible curve as $F=0$, with $F$ homogeneous of degree $n$. For $n=1$ there are no singular points. Otherwise choose a nonzero partial derivative, say $F_x$, which has degree $n-1$. Such a derivative exists in <characteristic zero>. Irreducibility and its smaller degree show that $F$ and $F_x$ have no common component.
At every <singular point> $p$ of $C$, $F_x(p)=0$ and the curve multiplicity of $F$ is at least two. The standard local lower bound for <intersection multiplicity> gives $I_p(F,F_x)\geq2$. Applying the <Bézout theorem>,
$$
2\#\operatorname{Sing}(C)\leq\sum_{p\in C\cap\{F_x=0\}}I_p(F,F_x)=n(n-1).
$$
Therefore
$$
\boxed{\#\operatorname{Sing}(C)\leq\tfrac12n(n-1).}
$$
This is the <polar bound for singular points of a plane curve>.
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