  Let $A$ be a connection of a principal bundle $P$ over a Riemannian manifold
$M$, such that its curvature $F_A\in L_{\text{loc}}^2(M)$ satisfies the
stationarity equation. It is a consequence of the stationarity that
$\theta_A(x,r)=e^{cr^2}r^{4-n}\int_{B_r(x)}|F_A|^2$ is monotonically increasing
in $r$, for some $c$ depending only on the local geometry of $M$. We are
interested in the singular set defined by $S(A)=\{x: \lim_{r\to
0}\theta_A(x,r)\neq 0\}$, and its stratification $S^k(A)=\{x: \text{no tangent
measure at $x$ is $(k+1)$-symmetric}\}$. We then introduce and study the
quantitative stratification $S^k_{\epsilon}(A)$. Roughly speaking,
$S^k_{\epsilon}(A)$ consists of points at which no tangent measure of $A$ is
$\epsilon$-close to being $(k+1)$-symmetric. In the main Theorem, we show that
$S^k_{\epsilon}$ is $k$-rectifiable and satisfies the Minkowski volume estimate
$\text{Vol}(B_r(S^k_{\epsilon})\cap B_1)\le Cr^{n-k}$. Lastly, we apply the
main theorems to the stationary Yang-Mills connections to obtain a
rectifiability theorem that extends some previously known results by G. Tian.
