Micro-or nano-structurally roughened solid surfaces exhibit a rich variety of wetting behavior types, ranging from superhydro- or superoleophobicity to superhydro- or superoleophilicity. Depending on their material chemistry, the scale and morphology of their roughness or even the application of external electric fields, their apparent wettability can be significantly modified giving rise to challenging technological applications by exploiting the associated capillary phenomena at the micrometer scale. Certain applications, however, are limited by hysteretic wetting transitions, which inhibit spontaneous switching between wetting states, requiring external stimuli or actuation like thermal heating. The presence of surface roughness, necessary for the manifestation of the superhydrophobicity, induces multiplicity of wetting states and the inevitable hysteresis appears due to considerable energy barriers separating the equilibrium states. Here, by using continuum as well as mesoscopic computational analysis we perform a systems level study of the mechanisms of wetting transitions on model structured solid surfaces. By tracing entire equilibrium solution families and determining their relative stability we are able to illuminate mechanisms of wetting transitions and compute the corresponding energy barriers. The implementation of our analysis to 'real world' structured or unstructured surfaces is straightforward, rendering our computational tools valuable not only for the realization of surfaces with addressable wettability through roughness design, but also for the design of suitable actuation for optimal switching between wetting states.