We describe the steady states, fluctuations, dynamics, and instabilities of atom and of vacancy single-layer-height islands during electromigration, assuming an isotropic medium. We emphasize the dependence on cluster size and on the three standard limiting cases of mass-transport mechanism: periphery diffusion (PD). terrace diffusion (TD). or evaporation-condensation (EC), as well as the differences between atom and vacancy clusters. A general model provides power laws describing the size dependence of the drift velocity in these limits, consistent with established tin the case of PD) results. For PD, atom and vacancy islands drift in opposite directions; otherwise they drift in the same direction. The validity of the widely used quasistatic limit is calculated. Linear stability analysis reveals a new type of morphological instability, not leading to island break-down. We find non-circular steady states for EC vacancy islands. Analytical calculations are corroborated by both Monte Carlo simulations and numerical integration. For weak electromigration the cluster responds isotropically for TD and PD but not EC. In EC, clusters elongate perpendicular to the drift axis. In PD a morphological instability at strong electromigration leads to cluster splitting, in contrast to destabilizing into slits in the other cases. TD or EC induces a new instability for vacancy clusters above a threshold. Using Langevin formalism, we derive the non-equilibrium cluster diffusion constant and study morphological fluctuations. Electromigration affects the diffusion coefficient of the cluster and morphological fluctuations, which diverge at the instability threshold. An intrinsic attachment-detachment bias displays the same scaling signature as PD in the drift velocity.