The mapping class group acts on the set of representations modulo conjugation of the fundamental group of n-punctured genus g-curves. For representations with values in SL(2,C), finite orbits of this action have been classified in the literature under various additional constraints. We complete this classification by the remaining case of reducible representations for g>0. This study is motivated by the following result: up to some minor technical conditions, representations modulo conjugation with values in GL(r,C) that have finite orbit under the action of the mapping class group are precisely those that appear as the monodromy of a logarithmic connection on the curve, with poles at the punctures, that admits an algebraic universal isomonodromic deformation. Both results concern a joint work with G. Cousin.