We examine the behavior of the first-order rainbow for a coated sphere by using both ray theory and Aden-Kerker wave theory as the radius of the core a 1 2 and the thickness of the coating b are varied. As the ratio b/a 12 increases from 10-4 to 0.33, we find three classes of rainbow phenomena that cannot occur for a homogeneous-sphere rainbow. For b/a12 < 10-3, the rainbow intensity is an oscillatory function of the coating thickness, for 8/a 1 2 10-2, the first-order rainbow breaks into a pair of twin rainbows, and for 8/a,2 0.33, various rainbow-extinction transitions occur. Each of these effects is analyzed, and their physical interpretations are given. A Debye series decomposition of coated-sphere partial-wave scattering amplitudes is also performed and aids in the analysis.