Motivated by a diversity of the shape instability phenomena in condensed-matter physics, we study the formation of elastic ribbon structures and transformation into helicoidal structures. Using the multipulse laser-matter interaction with the Co-coated surface, a one-dimensional high-density vortex-filament array has been created. Increasing the number of pulses, the oscillatory strain field causes the cascade of the shape transformations into structures of increasing topological complexity: vortex filaments into ribbons, into ribbon helicoids and tubular-ribbon helicoids, and then into short ribbon structures with the complex Scherk surface being identified. The cascade of transformations follows the scenario in which the topological complexity of the new structure increases with the number of pulses, thus, realizing configurations with more efficient energy relaxation. Above a critical number of pulses, the system is catastrophically disintegrated into small-scale wrinkled and crumpled surfaces. We show that the critical parameter for the ribbon transformations is the number of laser pulses which is equivalent to some critical oscillatory frequency.