For a topological space $X$ and a sheaf of abelian groups $F$ on it, sheaf cohomology $H^n(X,F)$ is defined.

Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally contractible and $F$ is the sheaf of locally constant functions.

I have two related questions.

For an algebraic scheme $X$, one uses the sheaf cohomology with the structure sheaf $F=\mathcal{O}_X$. What happens if $X$ is a topological manifold and $F$ is the sheaf of continuous functions to the real numbers? Or differentiable manifolds? Does this cohomology have a special name under which I can search for literature?

Doesn't one get many interesting cohomology theories besides singular cohomology for a topological space $X$ from sheaf cohomology? I mean for a classical topological space, not a scheme, which is Hausdorff and so forth. Is there a list or an overview in the literature? Thank you.

I like to restate the second question. Does any reasonable cohomology theory of topological spaces come from sheaf cohomology?