Classical semiparametric inference with missing outcome data is not robust to contamination of the observed data and a single observation can have arbitrarily large influence on estimation of a parameter of interest. This sensitivity is exacerbated when inverse probability weighting methods are used, which may overweight contaminated observations. Inverse probability weighted, double robust and outcome regression estimators of location and scale parameters are introduced, which are robust to contamination in the sense that their influence function is bounded. Asymptotic properties are deduced and finite sample behaviour studied. Simulated experiments show that contamination can be more serious a threat to the quality of inference than model misspecification. An interesting aspect of the results is that the auxiliary outcome model used to adjust for ignorable missingness by some of the estimators, is also useful to protect against contamination. Both adjustment to ignorable missingness and protection against contamination are achieved through weighting schemes. A case study illustrates how the resulting weights can be studied to gain insights on how the two different weighting schemes interact.