This is essentially a long comment in response to the other two answers.  

One place in which interesting homotopy types do appear is in the study of Hom complexes of graphs.  
<a href="http://arxiv.org/abs/math/0510177"> Csorba and Lutz </a> showed that Hom$(K_{2r} - C_{2r}, K_{r+1})$ is an orientable surface (not just up to homotopy, up to homeomorphism).  Here $C_k$ denotes a length $k$ cycle.  The genus is given by $r! \frac{r^2 - r -2}{2} + 1$, so it's never a sphere.  They list some other interesting conjectures and particular computations.

More recently, <a href="http://arxiv.org/abs/math/0510535"> Schultz </a> proved a conjecture of Csorba stating that Hom$(C_5, K_{n+2})$ is homeomorphic to the Stiefel manifold $V_2 (\mathbb{R}^{n+1})$ or orthnormal 2-frames.  This conjecture was made based on a complete calculation, by Kozlov, of the cohomology of Hom$(C_m, K_n); \mathbb{Z})$.  Surprisingly, these complexes have 2-torsion in their cohomology when n is even (are there other complexes arising in combinatorics that have torsion in their cohomology?).  Schutlz also showed that the colimit as $m\to \infty$ of the complexes `Hom$(C_{2m}, K_n)$ is homotopy equivalent to the free loop space on $S^{n-2}$!

So, these are some instances in which people worked hard to get interesting answers.

I guess it's also worth pointing out that for any finite simplicial complex X and any graph T, there is a graph G (with looped vertices) such that Hom$(T, G) \simeq X$.  This is a theorem of Anton Dochtermann.  One might argue that this violates the spirit of the question, since one can't really say that these Hom complexes appear naturally; instead the graphs G in some rough sense look like the space X you're trying to model.  (Specifically, G is the 1-skeleton of some subdivision of X, with loops placed on all the vertices.)