We present a general analytical approach to study the wrinkling phenomena in the Lame, sheet on drop, drop on sheet, and twisted ribbon setups. In the partially single-mode wrinkle pattern, we employ the boundary layer analysis to regularize the divergence in the energy and obtain the optimal wavenumber. In the multi-mode wrinkle pattern scenario, we consider the effects of the boundary condition, meniscus, and the geometry of the setup separately on the energies and obtain scaling results for the wavenumber of the multi-mode pattern due to each effect. Lastly, a wrinkle-to-fold transition is discussed in the Lame setup, with our main finding that this transition is triggered by geometry of the annulus alone which minimizes the surface energy