We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by Alberti and Müller (2001) we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on piecewise-linear periodic functions of slope +1 whose period depends on the location in the domain and the weights in the energy.