Fibrous surface structures can improve the adhesion of objects to other surfaces. Animals, such as flies and geckos, take advantage of this principle by developing "hairy" contact structures which ensure controlled and repeatable adhesion and detachment. Mathematical models for fiber adhesion predict pronounced dependencies of contact performance on the geometry and the elastic properties of the fibers. In this paper the limits of such contacts imposed by fiber strength, fiber condensation, compliance, and ideal contact strength are modeled for spherical contact tips. Based on this, we introduce the concept of "adhesion design maps" which visualize the predicted mechanical behavior. The maps are useful for understanding biological systems and for guiding experimentation to achieve optimum artificial contacts.