There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying some nice condition. Then $H^i(X, E)=0$ for $i>0$."

What is the best way to formulate this result? I ask because I'm sure I'll wind up teaching this material one day, and I'd like to get this right.

I asked a similar question over at nLab. Anyone who really understands this material might want to write something over there. If I come to be such a person, I'll do the writing!

Two versions I know:

(1) Suppose that, for any open cover $U_i$ of $X$, there are functions $f_i$ and open sets $V_i$ such that $\sum f_i=1$ and $\mathrm{Supp}(f_i) \subseteq U_i$. Then, for $E$ any sheaf of $\mathcal{O}_X$ modules, $H^i(X,E)=0$. Unravelling the definition of support, $\mathrm{Supp}(f_i) \subseteq U_i$ means that there exist open sets $V_i$ such that $X = U_i \cup V_i$ and $f_i|_{V_i}=0$.

Notice that the existence of partitions of unity is sometimes stated as the weaker condition that $f_i$ is zero on the closed set $X \setminus U_i$. If $X$ is regular, I believe the existence of partitions of unity in one sense implies the other. However, I care about algebraic geometry, and affine schemes have partitions of unity in the weak sense but not the strong.

(2) Any quasi-coherent sheaf on an affine scheme has no higher sheaf cohomology. (Hartshorne III.3.5 in the noetherian case; he cites EGA III.1.3.1 for the general case.) There is a similar result for the sheaf of analytic functions: see Cartan's Theorems.

I have some ideas about how this might generalize to locally ringed spaces other than schemes, but I am holding off because someone probably knows a better answer.

It looks like the answer I'm getting is "no one knows a criterion better than fine/soft sheaves." Thanks for all the help. I've written a blog post explaining why I think that fine sheaves aren't such a great answer on non-Hausdorff spaces like schemes.