I had a look to Nestruev's book a lot of time ago. I don't remember: didn't you need to consider closed (w.r.t. some -perhaps Fréchet- topology on $A$) maximal ideals to have the points of $M$?

First of all, what do you mean by "the spectrum of the $\mathbb{R}$-algebra $A$"? I don't think it's useful to take the prime spectrum in the sense of (algebraic) scheme theory. On the other hand, if you considered $C^{\infty}(M)$ as a $C^{*}$-algebra with trivial involution, perhaps you would get some compact Housdorff topological space different from $M$ ?...

Silly example: G Lie group, H closed subgroup, $G\rightarrow G/H$ is a (principal) fiber bundle whose total space (=G) is parallelizable. Can this picture be generalized a bit?

Just to make things simple, perhaps we could consider the case of principal bundles first, cause Lie groups are parallelizable manifolds...

I guess the question is: "When is the total space of a smooth fiber bundle a parallelizable manifold?"

Thanks for the clarification about punctures etc. - As for the pryms, is there a reference describing what happens over the "bad" points?

Edit: should add "semistable".

Ok. Let's stick to torsion-free sheaves of rank 2 and fixed determinant (in an algebro-geometric/holomorphic setting), if it can help to "restrict" the literature.

I need that the variety over which we consider these structures is a singular (projective) algebraic curve.

Also, in the article by Konno you linked, the base curve is a "marked Riemann surface", hence a smooth curve. So, it's kind of off topic with respect to this MO question.

In the page of Biquard I couldn't find articles about the topic I'm asking for (but perhaps I just didn't search carefully enough). Could you be more specific? There is an article about certain Higgs bundles on curves, but it seems to me that in that case the objects that are allowed to be singular are the Higgs fields (which are meromorphic) and the integrable connections, not the base curve. Plese correct me if I'm mistaken.

As far as I understand, the article by Nasatyr-Steer you linked is about orbifold curves (which are essentially smooth curves together with some additional data on a finite set of points), not to singular curves.

Could you please specify which article(s) of Simpson in the link actually refers to bundles over singular curves?

Thank you for the reference. But I think most of those articles are in the context of smooth curves, aren't they? Maybe the article of Faltings on bundles on semistable curves...