The robustness of the Lagrange Multiplier (LM) tests for spatial error dependence of Burridge (1980) and Born and Breitung (2011) for the linear regression model, and Anselin (1988) and Debarsy and Etur (2010) for the panel regression model with random or fixed effects are examined. While all tests are asymptotically robust against distributional mis-specification, their finite sample behaviour may be sensitive to the spatial layout. To overcome this shortcoming, standardized LM tests are suggested. Monte Carlo results show that the new tests possess good finite sample properties. An important observation made throughout this study is that the LM tests for spatial dependence need to be both mean- and variance-adjusted for good finite sample performance to be achieved. The former is, however, often neglected in the literature.