Morphological instabilities in electrodeposition have long been studied due to their important applications in electroplating and energy storage. They are receiving increased attention due to their prevalence in batteries with lithium metal anodes which are critical to next generation energy storage devices. These instabilities are driven by preferential charge transport across an electrolyte to the tips of perturbations to the metal surface. Using linear stability analysis, four models are developed to study the growth of morphological instabilities in electrodeposition under the action of various driving forces. The first model considers the ion transport across an electrolyte with fixed anions and demonstrates that spatial immobilization of anions can significantly reduce the electric field at the metal surface. The following model develops a framework to include the mechanical response of an elastic solid separator, which is then used to show that the combined effect of reduced electric field and elasticity-induced suppression can stabilize the deposition at all length scales. The next idea investigates the effect of an interfacial layer and yields that homogenization of cation concentration at the metal surface by lateral transport can weaken the growth of the instability. Finally, polymer additives are studied as a means to stabilize electroconvection-induced destabilization of the metal surface.