If current technology is used when wind turbines are located in deeper water, the cost of the foundations will increase dramatically. Thus, a new technology is needed to reduce the total cost of offshore wind turbines. A candidate technology could be the bucket foundation, also called a “suction caisson”. During installation, suction is applied inside the bucket. The suction creates a pressure differential across the bucket lid, effectively increasing the downward force on the bucket while the water flow reduces the skirt tip resistance. The bucket foundation is constructed as a thin shell structure. As the water depth increases, the forces from wind, waves, and currents also increase. Thus, a larger bucket diameter is required to sustain the larger moment. Hence, the aspect ratio between the bucket diameter and the skirt thickness becomes very large, and instability, in the form of buckling, becomes a crucial issue. Instability is also an issue affecting piles during installation, where imperfections and boulder impacts during pile driving can lead to structural buckling. The failure mechanism can be an “extrusion” mechanism type, where an initial imperfection in the pile geometry increases as the pile is penetrating through a soil of high stiffness. This paper deals with structural instability issues related to slender, circular cylindrical offshore foundations during installation. The authors present the previous work done by other authors within this field, highlighting the key issues, and proposing ideas for solutions.