Below, always let $A$ be the coordinate ring of a smooth affine variety over $\mathbb C$.
What can be said about the (non)-triviality of the module of Kahler differentials $\Omega_{A/\mathbb C}^1$?
Question: Is there an example of $A$ where $\Omega_{A/\mathbb C}^1$ is not free? Not stably free?
It seems like there must be an example of an affine variety with a non-trivial cotangent bundle, but I don't know it.
Taking a hypersurface in $\mathbb A^n$ will at best produce something stably free, and when $n=2$ it is always free, as I learned here. Still, it would be interesting to know an example where it is non-free.
If you work over $\mathbb R$, as in Swan's paper the $n$-sphere $A = \mathbb R[x_i]/(\sum x_i^2 -1)$ has a non-trivial tangent bundle (for $n\not= 1,3,7$), but still it is stably free and becomes free when passing to $\mathbb C$.
Update:
I have noticed that the question about parallelizability of hypersurfaces is addressed on overflow.
This is not so important for your question, but let me state it anyway. $\Omega^1_{A/k}$ is stably free if and only if it is a complete intersection in some embedding in the affine space (which in turn is equivalent to being a complete intersection in any large enough affine space). Having said that, take any smooth projective curve $C$ of genus $g\geq 2$ and take a general point $P$. Then $C-\{P\}$ is affine, but $\Omega^1$ is not free, since that is equivalent to saying $K_C=(2g-2)P$.