  In previous papers, we used the standard simplices $\Delta^p$ $(p\ge 0)$
endowed with diffeologies having several good properties to introduce the
singular complex $S^\dcal(X)$ of a diffeological space $X$. On the other hand,
Hector and Christensen-Wu used the standard simplices $\Delta^p_{\rm sub}$
$(p\ge 0)$ endowed with the sub-diffeology of $\rbb^{p+1}$ and the standard
affine $p$-spaces $\abb^p$ $(p\ge 0)$ to introduce the singular complexes
$S^\dcal_{\rm sub}(X)$ and $S^\dcal_{\rm aff}(X)$, respectively, of a
diffeological space $X$. In this paper, we prove that $S^\dcal(X)$ is a fibrant
approximation both of $S^\dcal_{\rm sub}(X)$ and $S^\dcal_{\rm aff}(X)$. This
result easily implies that the homotopy groups of $S^\dcal_{\rm sub}(X)$ and
$S^\dcal_{\rm aff}(X)$ are isomorphic to the smooth homotopy groups of $X$,
proving a conjecture of Christensen and Wu. Further, we characterize
diffeological principal bundles (i.e., principal bundles in the sense of
Iglesias-Zemmour) using the singular functor $S^\dcal_{\rm aff}$. By using
these results, we extend characteristic classes for $\dcal$-numerable principal
bundles to characteristic classes for diffeological principal bundles.
