Stable summands of $\Omega^kS^{n+k}$ The answer to this question might be known, but I don't know of any reference and will appreciate any references.
Let $n>0$. By Snaith splitting there is a stable splitting of $\Omega^kS^{n+k}$ into wedge of spaces $D_{k,r}S^n=F(\mathbb{R}^k,r)\ltimes_{\Sigma_r}(S^n)^{\wedge r}$ as
$$\Sigma^\infty\Omega^kS^{n+k}\simeq\bigvee_{r=1}^{+\infty}\Sigma^\infty D_{k,r}S^n$$
where $F(\mathbb{R}^k,r)$ is the configuaration space of $r$ distinct points in $\mathbb{R}^k$. 
It is known that $D_{k,2}S^n=\Sigma^nP_{n}^{n+k-1}$ where $P_n^m$ is the truncated real projective space $P^m/P^{n-1}$. 
What I would like to know is about other pieces and specially $D_{k,2^s}S^n$ with $s>0$. Is it known that for any positive $k,s,n$ the space $D_{k,2^s}S^n$ is suspension of some $CW$-complex? If the answer is positive, do we know how many suspension may appear?
 A: The full story is eluded to in the comments of Arone and Rognes.  First of all, $D_{k,r}\Sigma^n X$ is always an $n$-fold suspension.  More exciting is that, for all pairs $(k,r)$, there is a natural number $d = d(k,r)$, such that
$D_{k,r} \Sigma^dX = \Sigma^{rd}D_{k,r}X$.
So what is $d$? It is the order of the canonical $r$-dimensional vector bundle over the configuration space $F(\mathbb R^k, r)/\Sigma_r$.  This is a bundle of finite order because it has finite structure group and a finite dimensional base space.  The actual numbers $d(k,r)$ were computed a long time ago:  see [Cohen, Cohen, Kuhn, Neisendorfer, Bundles over configuration spaces, Pac. J. Math. 104 (1983), 47-54].  And yes, $d(2,r)=2$ for all $r$, was known even earlier.
A: When $k=2$ and $n$ is even,
$$
D_{k,r} S^n \simeq BB_{r+} \wedge S^{rn}
$$
is the $rn$-fold unreduced suspension of the classifying space of the braid group $B_r$ on $r$ strings. I once cited Cohen-Mahowald-Milgram ``The stable decomposition for the double loop space of a sphere'' (1978) for this, and if I recall correctly they credited Arnold for the idea.  Hence $rn$ is the maximal number of suspensions that can appear in this case.
Edit: For $k=2$ and $n=2$ the map $F(\mathbb{R}^2,r) \times_{\Sigma_r} \mathbb{R}^{2r} \to \mathbb{R}^{2r}$ taking $[z,\xi] = [z_1, \dots, z_r, \xi_1, \dots, \xi_r]$ to
$$
(\sum_i \xi_i, \sum_i z_i \xi_i, \dots, \sum_i z_i^{r-1} \xi_i)
$$
trivializes the source as an $\mathbb{R}^{2r}$-bundle over $F(\mathbb{R}^2,r)/\Sigma_r$.  Here the $z_i$ and $\xi_i$ in $\mathbb{R}^2$ are viewed as complex numbers, and the products in the displayed formula are formed in $\mathbb{C}$.  A similar formula works for any even $n$, by taking the direct sum of $n/2$ copies of this $\mathbb{R}^{2r}$-bundle.  Hence the Thom complex of this bundle, which is your $D_{2,r} S^n$, is the $rn$-th suspension of $(F(\mathbb{R}^2,r)/\Sigma_r)_+ \simeq BB_{r+}$.  I recalled this in Proposition 9.10 of my paper "Topological logarithmic structures" GTM 16 (2009).
For $k=2$ and $n=1$, or more generally for $n$ odd, the stable splitting of $\Omega^2 S^3$ is closely connected to the story of Brown-Gitler spectra.
Regarding the case when $k$ is a higher power of $2$, you might first try $k=4$ and  identify $\mathbb{R}^4$ with the quaternions.  However, in that case the quaternionic Vandermonde matrix might well be singular.  For instance, with $r=3$ and $z = (i,j,k)$ the matrix
$\begin{pmatrix} 1 & 1 & 1 \\ i & j & k \\ -1 & -1 & -1 \end{pmatrix}$
annihilates $\xi = (1,0,1)$, so Arnold's idea does not extend to this case.
