In this article we consider the connected component of the identity ofG-character varieties of compact Riemann surfaces of genusg>0 , for connected complex reductive groups G of type A (e.g., SLn and GLn). We show that these varieties are Q-factorial symplectic singularities and classify which admit symplectic resolutions. The classification reduces to the semi-simple case, where we show that a resolution exists if and only if either g=1 and G is a product of special linear groups of any rank and copies of the group PGL2, or ifg=2 and G=(SL2)m for some m.