Let $n>1$. The homology of the free loop space $\Lambda S^n$ of the sphere $S^n$ contains two torsion if $n$ is even. Thus the fibration $$ \Omega S^n\rightarrow \Lambda S^n\rightarrow S^n $$ is not trivial if $n$ is even (Here $\Omega S^n$ denotes the based loop space.).

The odd dimensional spheres do not have torsion in the homology of the free loop space and one can compute that $H_*(\Lambda S^{2k+1};\mathbb Z)\cong H_*(\Omega S^{2k+1}\times S^{2k+1};\mathbb Z)$. Hence the homology does not obstruct the existence of a trivialization of the free loop fibration. Indeed $\Lambda S^3$ and $\Omega S^3\times S^3$ are homeomorphic, which can be proven using the group structure on $S^3$.

Is the free loop space fibration always trivial for odd dimensional spheres ? My guess would be that this is not the case, with a possible exception of $S^7$.