What is the structure of the space of solutions of a non linear ODE? As is well known, the space of solutions of a linear ODE with, say, $\mathcal{C}^\infty$ coefficients on $\mathbb{R}$ is a finite dimensional affine space (a vector space, in the homogeneus case).

What is the structure of the space of solutions of a non linear ODE? In which cases does it have a "natural" structure of finite dimensional smooth manifold? 

For the question to make sense, some conditions on the form of the equation must be put. Let's assume the ODE is of the form
$$F(t,u(t),u'(t),u''(t),\dots,u^{(n)}(t))=0$$
where $F:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}$ is a smooth function. 
 A: Deep and interesting theory exists when $F$ is analytic, not just smooth. 
It was developed by J. F. Ritt in his books on Differential algebra.
(In particular, Ritt rigorously defined the things like "singular solutions").
Some parts of the theory also apply to PDE.
Most interesting and useful theory is obtained when $F$ is a polynomial,
in which case the set of solution is a "differential-algebraic manifold",
and a theory of DA manifolds is parallel to the theory of the usual algebraic manifolds.
For a more modern exposition (including fields of non-zero characteristic)
see Kolchin, Differential algebra and algebraic groups.
A: There is yet another direction in which one could take the question, one that makes more contact with the linear and affine structures that one observes in the standard treatment of linear equations (both homogeneous and inhomogeneous).  For example, consider the Riccati equation
$$
u'(t) = a_0(t) + a_1(t)u(t) + a_2(t)u(t)^2,\tag{1}
$$
which is nonlinear (when $a_2$ is not zero).  A standard fact that is proved in ODE courses is that there is a 'trick' for 'linearizing' this equation, yielding the result that the general solution can be written in the form
$$
u(t) = \frac{c_1b_1(t)+c_2b_2(t)}{c_1b_3(t)+c_2b_4(t)}
$$
for some functions $b_i(t)$ (defined on the entire interval on which the $a_i$ are defined) and constants $c_1$ and $c_2$ (not both zero).  Obviously, only the ratio of $c_1$ to $c_2$ matters for specifying $u$, so the ($1$-dimensional) space of solutions to $(1)$ is 'naturally' diffeomorphic to the real projective line $\mathbb{RP}^1$.  Note that this representation works even when the solution $u$ has a 'pole'.  (For example, consider the Riccati equation $u' = 1 + u^2$, which has $u(t) = \tan t$ as a solution.)  
Sophus Lie developed a theory that associated a type of ODE to each homogeneous space of what we now call Lie groups, one that generalized the theory of linear ODE (both homogeneous and inhomogeneous) and Riccati equations in their various forms (which correspond to vector spaces, affine spaces, and projective spaces, thought of as homogeneous spaces of appropriate Lie groups).  
It's possible that the OP is interested in learning more about the theory of ODE (and PDE) of Lie type, which is what this theory developed into.  For this theory, putting a (possibly non-Hausdorff) manifold structure on the space of solutions is a necessary first step, but it is only the beginning.
A: As Robert Bryant observed, something like the solubility of $F$ for $u^{(n)}$ needs to be assumed. Then by the trick (due I believe to D'Alembert) of setting $x=(t,u,\dots,u^{(n-1)})$ and dt/ds=1, the equation can always be rewritten
$$
\frac{dx}{ds}=f(x).
$$
This is what most geometers would call the "standard ODE", wherein $f$ is a smooth vector field on the manifold where $x$ evolves.
In this setting, the space of (maximal connected) solution curves is indeed always a (not necessarily Hausdorff) manifold. I don't know who first wrote this, but according to this recent obituary a contender would be J.-M. Souriau in his 1970 book "Structure des systèmes dynamiques". Translation of the relevant passage:

The manifold of motions of a system
Jean-Marie Souriau has observed that the set of maximal solutions of a differential system on a differentiable manifold possesses itself a natural structure of differentiable manifold, not always separated: thus one can speak of the system's manifold of motions. This very simple property is rarely mentioned in the usual courses on differential calculus.  

