The inferential problem of drawing inference about a common mean µ of several independent normal populations with unequal variances has drawn universal attention, and there are many exact and asymptotic tests for testing a null hypothesis H₀ : µ = µ₀ against two-sided alternatives. In this paper we provide a review of some of these exact and asymptotic tests, and present theoretical expressions of local powers of the exact tests and a comparison. It turns out that, in the case of equal sample size, a uniform comparison and ordering of the exact tests based on their local power can be carried out even when the variances are unknown. Our observation is that both modified F and modified t tests based on a suitable combination of component F and t statistics perform the best in terms of local power among all exact tests under consideration. An exact test based on inverse normal method of combination of P-values also performs reasonably well.