Analogue of Quillen-Suslin theorem for affine varieties Phrased in the language of vector bundles, the Quillen-Suslin theorem states that vector bundles on $\mathbb C^n$ are algebraically trivial (for any algebraic vector bundle there exists an algebraic isomorphism to the trivial bundle).
For more general affine varieties, Grauert's Oka principle implies that the holomorphic and topological classification of vector bundles coincide. In particular, all algebraic vector bundles which are topologically trivial are also holomorphically trivial.
As far as I understand, it is not known whether topologically trivial algebraic bundles on affine varieties are algebraically isomorphic to the trivial bundle.
If this would be the case for an affine variety $X$, I would call this an analogue of the Quillen-Suslin theorem for $X$.
My question is whether such analogues of the Quillen-Suslin theorem have been proved for affine varieties other than $\mathbb C^n$.
 A: More of an aside than anything, but we will need to distinguish between holomorphic and algebraic Quillen Suslin statements. Grauert's Oka principle can show a vector bundle is holomorphically trivial but on affine varieties holomorphically trivial doesn't imply algebraically trivial.
Expanding on the comments of Starr and Tosteson to illustrate this subtlety: when $X = Y \setminus \{p\}$ is the
complement of a point on a smooth projective curve $Y$ of genus $g>
0$, $\mathrm{Pic}(X) \simeq \mathrm{Pic}(Y) / \mathbb{Z} \simeq
\mathrm{Pic}^{0}(Y)$ where the last isomorphism is obtained from
$\mathrm{Pic}(Y) \simeq \mathrm{Pic}^{0}(Y) \times \mathbb{Z}$ (an
exercise in Hartshorne II.6). On the other hand it is true that all line bundles on $X$ are holomorphically trivial: this can be seen from the exponential exact sequence 
$$
\cdots \to H^1(X, \mathcal{O}_X) \to H^1(X, \mathcal{O}_X^\times) \to  H^2(X, \mathbb{Z}) \to  H^2(X, \mathcal{O}_X) \to \cdots
$$
And vanishing for higher cohomology of $\mathcal{O}_X$ (Cartan AB) and $ H^2(X, \mathbb{Z})$ $X$ is homotopy equivalent to a wedge of $2g$ circles).
A neccessary conditon
If $X$ is a smooth variety and every algebraic vector bundle on $X$ is
trivial, then $K_{0}(X) = \mathbb{Z}$, generated by the class of
$\mathcal{O}_{X}$, a severe restriction, for instance after tensoring
with $\mathbb{Q}$ we have $\mathrm{CH}^{*}(X) \otimes \mathbb{Q} \simeq
K_{0}(X) \otimes \mathbb{Q} = \mathbb{Q}$.
In the example above a key point was $\mathrm{CH}^*(X) \not \simeq \mathbb{Q}$.
One answer to the question
A conjecture of Anderson/theorem of Gubeladze shows that if $X$ is an
affine toric variety, then every vector bundle on $X$ is free. The primary source is 
I. Gubeladze, The Anderson conjecture and projective modules over monoid algebras, Soobshch. Akad. Nauk Gruzin. SSR 125 (1987), 289–291
