= Solution
In this context a <nowhere locally homogeneous metric>, also called a bumpy metric, is a smooth <Riemannian metric> for which no two distinct nonempty open subsets are isometric with their induced metrics. Equivalently every <local isometry> between open subsets is the identity wherever defined: a nonidentity <local isometry> sends some point $x$ to a distinct point, and restriction to sufficiently small disjoint neighborhoods would violate the first formulation. This is the local-isometry meaning of the terminology here; degeneracy of periodic <geodesics> is a different use of the word bumpy.
<Sunada's local isometry lemma> states that on a compact smooth manifold without boundary of dimension $m\ge2$ the <nowhere locally homogeneous metrics> contain a <residual set> in the space of smooth <Riemannian metrics> with its $C^\infty$ topology. In particular they are dense, by the <Baire category theorem>. A <residual set> is a countable intersection of open dense sets. The dimension assumption matters: every one-dimensional <Riemannian metric> is locally $ds^2$ in arclength coordinates and has local translations.
The <jet bundle of maps> $J^k(M,N)$ consists of equivalence classes $j_x^kf$ of smooth maps near $x$, where two maps agree to order $k$ at $x$ in coordinate charts. Its projection to $M\times N$ sends $j_x^kf$ to $(x,f(x))$. For a <multi-index> $\alpha$ in $m$ variables there are $\binom{m+r-1}{r}$ derivatives of order $r$. Thus a coordinate chart consists of the source point, the target point, and $n$ coefficients for every derivative order $1$ through $k$. Summing the counts gives
$$
\boxed{\dim J^k(M,N)=m+n\binom{m+k}{k},\qquad
\dim J^k_{x,y}(M,N)=n\left(\binom{m+k}{k}-1\right).}
$$
This includes $J^0(M,N)=M\times N$. The fibre over $(x,y)$ is the space of truncated Taylor maps with fixed constant term $y$, locally modeled on
$$
\bigoplus_{r=1}^k\operatorname{Sym}^r(T_x^*M)\otimes T_yN.
$$
For $k=1$ its identification with $\operatorname{Hom}(T_xM,T_yN)$ is canonical. For higher $k$, changes of target coordinates mix derivatives of different orders, so this is a coordinate or connection-dependent description, not a canonical vector-bundle identification. The truncation $J^k\to J^{k-1}$ is an <affine bundle> modeled on the pullback of $\operatorname{Sym}^k(T^*M)\otimes TN$ over $M\times N$.
For the density assertion put $A=\overline U_i$ and $B=\overline U_j$. \b[Their closures must be distinct]. If $A=B$, the identity is an <isometry> for every <Riemannian metric>, and the requested complement is empty. Under the intended distinct-domain assumption, the smooth closed-ball hypothesis makes $A$ and $B$ regular closed domains, equal to the closures of their interiors. After interchanging them if necessary, there is a nonempty open set $W$ with compact closure in $\operatorname{int}(A)\setminus B$. Otherwise each interior would be contained in the other closure, forcing $A=B$.
Take any smooth <Riemannian metric> $g$. If $\operatorname{vol}_g(A)\ne\operatorname{vol}_g(B)$, it already lies outside $\mathcal S_{ij}$, because an <isometry> preserves the <Riemannian volume form>. If the volumes agree, choose a nonzero nonnegative smooth <bump function> $\eta$ supported in $W$ and set
$$
g_t=(1+t\eta)g\qquad(t>0).
$$
These are positive definite <Riemannian metrics>, they agree with $g$ on $B$, and $g_t\to g$ in $C^\infty$ as $t\downarrow0$, since every derivative of $g_t-g$ is $t$ times a fixed compactly supported smooth tensor. Their <Riemannian volume forms> satisfy
$$
dV_{g_t}=(1+t\eta)^{m/2}dV_g.
$$
Hence $\operatorname{vol}_{g_t}(B)=\operatorname{vol}_g(B)$ whereas $\operatorname{vol}_{g_t}(A)>\operatorname{vol}_g(A)$ for every $t>0$. In particular
$$
\left.\frac{d}{dt}\right|_{t=0}\operatorname{vol}_{g_t}(A)=\frac m2\int_A\eta\,dV_g>0.
$$
The two domains cannot be isometric for $g_t$. Every $C^\infty$ neighborhood of $g$ therefore meets the complement of $\mathcal S_{ij}$, proving
$$
\boxed{\overline{\mathcal C\mathcal S_{ij}}=\operatorname{Met}^{\infty}(M)\quad\text{if }\overline U_i\ne\overline U_j.}
$$
This <localized volume perturbation> proves the requested density even when the two domains overlap, and works in every positive dimension. It does not by itself prove the stronger residual local-isometry statement: fixed isometric closures and arbitrary isometric open subsets are different conditions.
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