The main purpose of this article is to present several new statistical tests of neutrality of mutations against a class of alternative models, under which DNA polymorphisms tend to exhibit excesses of rare alleles or young mutations. Another purpose is to study the powers of existing and newly developed tests and to examine the detailed pattern of polymorphisms under population growth, genetic hitchhiking and background selection. It is found that the polymorphic patterns in a DNA sample under logistic population growth and genetic hitchhiking are very similar and that one of the newly developed tests, F(s), is considerably more powerful than existing tests for rejecting the hypothesis of neutrality of mutations. Background selection gives rise to quite different polymorphic patterns than does logistic population growth or genetic hitchhiking, although all of them show excesses of rare alleles or young mutations. We show that Fu and Li's tests are among the most powerful tests against background selection. Implications of these results are discussed.