The H-test is known to be powerful against a wide range of possible pulse shapes. In this paper we derive analytic expressions for flux limits for single pulses (with any duty cycle) following a nondetection from this test (see Thompson et al. 1994 for an application of these results to the EGRET pulsar survey). General flux limits are also derived for Crab-, Vela-, and Geminga-type pulse profiles, which show that there is an experimental bias against Geminga-type profiles (the well-known Z(m)(2) test also shares this problem) which have a separation of 180 degrees. A few pulsars are known to emit phase-locked single pulses with a broadband nonthermal spectrum. The pulse phase mu has been determined for each of these pulsars, and future searches for pulsed emission at phase mu from any of these pulsars can be made with our proposed C-m or ''cosine'' test, which is shown to be more powerful than any previously known test. The significance of a detection is given by the value of C-m, which is N(0, 1) distributed for noise. Flux calculations and limits follow explicitly from Gaussian statistics. A consistency check (also in terms of Gaussian statistics) for phase locking is also derived for this test.
