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.
Csorba and Lutz 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, Schultz 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.)