We present radio transient search algorithms, results, and statistics from the ongoing Arecibo Pulsar ALFA (PALFA) Survey of the Galactic plane. We have discovered seven objects by detecting isolated dispersed pulses and one of the new discoveries has a duty cycle of 0.01%, the smallest known. The impact of selection effects on the detectability and classification of intermittent sources is discussed, and the relative efficiencies of periodicity vs. single pulse searches are compared for various pulsar classes. We find that scintillation, off-axis detection and few rotation periods within an observation may misrepresent normal periodic pulsars as intermittent sources. Finally, we derive constraints on transient pulse rate and flux density from the PALFA survey parameters and results. ; Comment: Added references, minor changes to Section 5.1. Accepted to ApJ