Storage ring lattice design is a highly constrained multiobjective optimization problem. The objectives can include lattice functions or derived quantities like emittance, brightness, or luminosity while simultaneously fulfilling constraints such as linear stability of the lattice. In this paper we explore the use of multiobjective genetic algorithms (MOGA) to find globally optimized lattice settings in a storage ring. Using the Advanced Light Source (ALS) for illustration, three examples of MOGA are shown and analyzed-(i) using three fit parameters to optimize the straight section betatron function and the natural emittance, (ii) using three fit parameters to optimize the photon brightness of bending magnet and insertion device source points in the lattice and (iii) a six parameter fit creating alternating high and low horizontal betatron functions in subsequent straight sections while still minimizing the natural emittance. Making use of one of the main benefits. of MOGA, we also study the trade-offs in the optimization objectives between sets of optimal solutions.