Many commonly used test statistics are based on a norm measuring the evidence against the null hypothesis. To understand how the choice of a norm affects power properties of tests in high dimensions, we study the consistency sets of $p$-norm based tests in the prototypical framework of sequence models with unrestricted parameter spaces, the null hypothesis being that all observations have zero mean. The consistency set of a test is here defined as the set of all arrays of alternatives the test is consistent against as the dimension of the parameter space diverges. We characterize the consistency sets of $p$-norm based tests and find, in particular, that the consistency against an array of alternatives cannot be determined solely in terms of the $p$-norm of the alternative. Our characterization also reveals an unexpected monotonicity result: namely that the consistency set is strictly increasing in $p \in (0, \infty)$, such that tests based on higher $p$ strictly dominate those based on lower $p$ in terms of consistency. This monotonicity allows us to construct novel tests that dominate, with respect to their consistency behavior, all $p$-norm based tests without sacrificing size.