How to find the smallest flabby sheaf containing a given sheaf? None of the spaces $C^k(\mathbb{R}^n)$, with $0 \leq k \leq \infty$, is a flabby sheaf. However, they are respectively contained in the smallest flabby sheaves $C^k_{nd} (\mathbb{R}^n)$ of functions $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ for which there exist $\Gamma \subset \mathbb{R}^n$, $\Gamma$ closed and nowhere dense, such that $f$ restricted to $\mathbb{R}^n \setminus \Gamma$ is $C^k$-smooth.
Is any similar way known to construct flabby sheaves which contain given sheaves of functions defined on some topological space $X$?
 A: There is the "flabbification" or "flasquification" functor used in the Godement resolution.  Namely, given a sheaf $\mathcal{F}$ on $X$, let $\mathcal{F}_x$ be the stalk at a point $x$.  Then we define the sheaf $\Phi(\mathcal{F})$ to have sections $$\Gamma(U, \Phi(\mathcal{F}))=\prod_{x\in U} \mathcal{F}_x$$
with the obvious restriction maps.  (This is the same as endowing the etale space of $\mathcal{F}$ with the trivial topology and considering the sheaf of sections.)  There is a natural injection $\mathcal{F}\to \Phi(\mathcal{F})$ sending a section of $\mathcal{F}$ to its stalks.  Note that $\Phi$ is functorial; indeed, it is the right adjoint to the natural inclusion of the full subcategory (Flasque sheaves on $X$) $\hookrightarrow$ (Sheaves on $X$).
See for example this Wikipedia article or Godement's book on sheaf theory.
Edit:  One can also mimic your construction of $C^k_{nd}$ as follows.  Let $\mathcal{F}_{nd}$ have global sections given by $\bigcup \Gamma(X-\partial U, \mathcal{F})/\sim$ where the union is taken over all open sets $U$, and we say $(f, X-\partial U)\sim (f', X-\partial U')$ if $f=f'$ when restricted to $X-(\partial U\cup \partial U')$.  Then local sections will be restrictions of these global sections.  This will always be flabby, but will not necessarily have a morphism $\mathcal{F}\to \mathcal{F}_{nd}$ unless $\mathcal{F}$ has enough sections (for example, if $\mathcal{F}$ is fine, as in your example).
