Smooth spline surfaces can now be built with polyhedral control nets rather than just grid-like tensor-product control nets. However, irregularities such as T-junctions, multi-sided facets, and n-valent vertices need to be sufficiently separated. Automatically generated quad-dominant meshes, and meshes created by designers unaware of the requirements for spline surfaces often pack irregularities too tightly.Global refinement, e.g. via two steps of subdivision, can sufficiently separate irregularities. However, each refinement quadruples the number of polynomial pieces. Moreover, first-step artifacts can lead to oscillating and pinched highlight line distributions. We therefore investigate minimal, single edge insertion, re-connection and localized refinement of quad-dominant meshes to make them suitable for polyhedral splines.