Limits of trivial bundles on curves

We extend the work of A. Beauville on rank 2 vector bundles on a smooth curve in several directions. We give families of examples with large dimension, add new existence and non-existence results and prove the existence of indecomposable limits with arbitrary rank. To construct the large dimensional families we use the examples of limits of rank 2 trivial bundles on P2 and P3 due to C. Banica. We also consider a more flexible notion: limits of trivial bundles on nearby curves.