Well, the MSE is another name for the residual variance estimate. Note that not all models which can be fit using the MIXED procedure have a simple residual variance estimate. If you have fitted a multivariate normal distribution (using a REPEATED statement), then you could have a residual variance for each response. How residual variance(s) is(/are) identified in a specific model depends on the variance structure which you are estimating. So, there is not a simple, direct answer to your question.
Does this circuitous answer help you to identify where to find the statistic in your output?