This paper develops a test for coefficient stability in spatial regressions. The test is designed to have good power for a wide range of persistent patterns of coefficient variation, be applicable in a wide range of spatial designs, and to accommodate both spatial correlation and spatial heteroskedasticity in regressors and regression errors. The test approximates the best local invariant test for coefficient stability in a Gaussian regression model with L´evy-Brown motion coefficient variation under the alternative, and is thus a spatial generalization of the Nyblom (1989) test of coefficient stability in time series regressions. An application to 1514 zip-code level bivariate regressions of U.S. socioeconomic variables reveals widespread coefficient instability.