It's hard to answer this question in detail in all generality. But we can at least give some high-level pointers here.
First of all, everything depends on the parameter you want to test, that is, AR, AAR, CAR, or CAAR. It is paramount to use a test statistic in the relevant category. As an analogy, the best method to cook a steak will yield unsatisfactory results if you are in the mood for pizza or sushi.
Within any category, we offer a sub-menu of test statistics. The main distinction here is between parametric and nonparametric tests, which is addressed in a separate question. in a nutshell, if the sample is small, a nonparametric test is always preferred, but even for large(r) sample sizes, nonparametric tests are not necessarily worse, although they (still) tend to be less used than parametric tests.
Last but not least, for any test to be valid (or trustworthy) a certain list of assumptions needs to be fulfilled. As an analogy, if you want to cook a certain recipe, you need to make sure that you have the proper ingredients and cooking equipment at hand; the best recipe for cooking a steak will fail if your main ingredient is a shoe sole instead. To learn more about this important topic, see the separate article "Which assumptions do the various test statistics make?".