Although many galaxies in the Virgo cluster are known to have lost significant amounts of HI gas, only about a dozen features are known where the HI extends significantly outside its parent galaxy. Previous numerical simulations have predicted that HI removed by ram pressure stripping should have column densities far in excess of the sensitivity limits of observational surveys. We construct a simple model to try and quantify how many streams we might expect to detect. This accounts for the expected random orientation of the streams in position and velocity space as well as the expected stream length and mass of stripped HI. Using archival data from the Arecibo Galaxy Environment Survey, we search for any streams which might previously have been missed in earlier analyses. We report the confident detection of ten streams as well as sixteen other less sure detections. We show that these well-match our analytic predictions for which galaxies should be actively losing gas, however the mass of the streams is typically far below the amount of missing HI in their parent galaxies, implying that a phase change and/or dispersal renders the gas undetectable. By estimating the orbital timescales we estimate that dissolution rates of 1-10 Msolar/yr are able to explain both the presence of a few long, massive streams and the greater number of shorter, less massive features.