The plane strain elasticity problem for a functionally gradient material (FGM) and a multi-layered homogeneous coating bonded to a metal substrate due to a uniform temperature change is considered. The substrate is assumed to be a nickel-based alloy-Rene-41. The FGM coating is a particulate composite of Rene-41 and zirconia with volume fractions continuously varying through the thickness. The multi-layered coating consists of one, two or four homogeneous layers with stepwise changing volume fractions. With a possible failure mechanism of surface cracking or edge debonding in mind, the relevant stresses on the surface of the coating and along the interfaces are evaluated. For the piecewise homogeneous coating the power of singularity and the corresponding stress intensity factors at the points of intersection of the interfaces and the free ends are calculated. It is shown that by using the FGM coating the stress singularities are eliminated and the stress distribution is smoothed considerably. Sample results for the displacements and for the contour plots of an effective stress governing the yield behavior of the material are given and the asymptotic behavior of the stress state around the singular point is examined.