Let $Y$ be a closed subvariety of a variety $X$ and let $i=Y\hookrightarrow X$. The normal sheaf of $Y$ in $X$ is the sheaf $$ \mathcal{N}_{Y/X}=(\mathscr I_Y/\mathscr I_Y^2)^\vee=\mathcal{Hom}_{\mathscr{O}_Y}(\mathscr I_Y/\mathscr I_Y^2,\mathscr{O}_Y) $$ I am specifically interested in the simple case $Y=\{\mathrm{pt}\}$ and $X=\Bbb C^r$ and I want to compute the sheaf cohomology $$ H^p(Y,\wedge^q\mathcal N_{Y/X})=H^p(\mathrm{pt},\wedge^q\mathcal N_{\mathrm{pt}/\Bbb C^r})\tag{1} $$ for all $p$ and $q$.
Since $Y$ is clearly affine we know that (1) is trivial for $p\neq 0$. Thus we want to compute the global sections of $\wedge^q\mathcal N_{\mathrm{pt}/\Bbb C^r}$ but I'm not sure what these are.
How can I compute these groups?
In the affine case, by Hartshorne, Corollary II 5.5, any coherent sheaf is of the form $M^\sim$ and the global sections are just $M$. So in your case, the global sections are just the $q$-th exterior power of the dual space of $M/M^2$, where $M$ denotes the maximal ideal corresponding to your point. The dimension can be computed with the binomial coefficient $\binom{r}{q}$.