We present the search methods and initial results for transient radio signals in the High Time Resolution Universe (HTRU) survey. The HTRU survey's single-pulse search, the software designed to perform the search and a determination of the HTRU survey's sensitivity to single pulses are described. Initial processing of a small fraction of the survey has produced 11 discoveries, all of which are sparsely emitting neutron stars, as well as provided confirmation of two previously unconfirmed neutron stars. Most of the newly discovered objects lie in regions surveyed previously, indicating both the improved sensitivity of the HTRU survey observing system and the dynamic nature of the radio sky. The cycles of active and null states in nulling pulsars, rotating radio transients (RRATs) and long-term intermittent pulsars are explored in the context of determining the relationship between these populations and of the sensitivity of a search to the various radio-intermittent neutron star populations. This analysis supports the case that many RRATs are in fact high-null-fraction pulsars (i.e. with a null fraction of greater than or similar to 0.95) and indicates that intermittent pulsars appear distinct from nulling pulsars in their activity cycle time-scales. We find that in the measured population, there is a deficit of pulsars with typical emission time-scales greater than similar to 300 s that is not readily explained by selection effects. The HTRU low-latitude survey will be capable of addressing whether this deficit is physical. We predict that the HTRU survey will explore pulsars with a broad range of nulling fractions (up to and beyond 0.999), and at its completion is likely to increase the currently known RRATs by a factor of more than 2.