Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.


1) Under what sufficient conditions on $F$ for any compact subset $K\subset X$ one has
$$R^i\Gamma\_K(X,F)=0\mbox{ for all } i>0,\,\,\,\,(1)$$
where $\Gamma\_K$ is the functor of global sections supported on $K$, and $R^i\Gamma\_K$ are its right derived functors?


2) Under what sufficient conditions on $F$ one has
$$R^i\Gamma\_{\{x\}}(X,F)=0\mbox{ for all } i>0,\,\,\,(2)$$
for any point $x\in X$.


3) Let $F$ be the sheaf of generalized functions on $X$. Is it true that either (1) or (2) are satisfied for this particular $F$?



Vanishing of sheaf cohomology with compact support
To complement Sándor's answer, let me focus on (3) since that seems to be what you are interested in. Let $F$ be the sheaf of generalized functions or distributions. Consider the localization sequence
$$ H^0(X,F)\to H^0(X-x,F)\to H\_x^1(X, F)\to H^1(X,F)$$
To ensure vanishing of the 3rd term, you would need to know that any distribution on $X-x$ extends to $X$. But surely this isn't true. (E.g. take $1/|x|$ viewed as a distribution on $X-x=\mathbb{R}-\{0\}$ and try to extend.; see Igor's comment, below)
 On the plus side, since $F$ is soft, the continuation of the same sequence shows that $H^i\_x(X, F)=0$ when $i\ge 2$.


