Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer by layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.