Let $C \subseteq \mathbb{P}^2$ be an irreducible plane curve of degree $d$ and let $p \in C$. For any integer $n \geq 1$, let $V_n$ be the set of all forms $f$ of degree $d1$, such that $f$ vanishes in $p$ and such that the intersection multiplicity of $C$ and the curve defined by $f$ is at least $n$. If $p$ is a smooth point of $C$, $V_n$ is a vector space. Is this also the case, if $p$ is a singularity of $C$? In particular, I am interested in the case, where the ground field are the complex numbers.
No. We might as well work in the affine plane for simplicity, since intersection multiplicity is defined locally. Let $C$ be defined by the equation $xy$. and let $P$ be the point $x=y=0$. Let $f_1=x+y^{n1}$, $f_2=y+x^{n1}$. These have intersection multiplicty $n$ with $C$, which you can compute by adding their intersection multiplicity with the curve defined by $x$ to their intersection multiplicity with the curve defined by $y$. But $f_1+f_2=x+y+x^{n1}+y^{n1}$, which has an intersection multiplicity of two. Edit: To make $C$ a curve of degree $n$, we multiply by any polynomial of degree $n2$ that that does not vanish at the origin. 


No, it is not always a vector space. Take $C$ a cubic with a node at $p$ and $n=3$. Then your set $V_3$ is a union of two vector spaces each corresponding to the conics that pass through $p$ and having as tangent at $p$ the tangent to one of the two branches to $C$ at $p$. The intersection of these two vector spaces is the set of multiples of the product of these two lines. 


If $C$ is unibranch at $p$, $V_n$ is always a vector space. If $C$ is not unibranch, there exist $n$ such that $V_n$ is not a vector space (as noted by the previous answers). To see the first claim, consider the normalization $\eta:\tilde C \rightarrow C$ of the curve. $C$ is unibranch at $p$ if $\eta^{1}(p)$ is a single point $q$. Let $x,y$ be affine coordinates in a neighborhood of $p$, and $t$ a parameter of $\tilde C$ at $q$. Consider the induced map $\eta^*:\mathbb{C}[x,y]\rightarrow \mathbb{C}[[t]]$ and the ideal $I_n=(t^n)\subset \mathbb{C}[[t]]$. Then $V_n$ can be identified with $(\eta^*)^{1}(I_n)\cap \mathbb{C}[x,y]_{\le d1},$ where $\mathbb{C}[x,y]_{\le d1}$ denotes the set of polynomials of degree at most $d1$. 

