As Bart noted, the primary benefits of the dual-pol upgrade for most of  the public will be better accumulated precipitation estimates (R-Kdp or R-Zdr/Kdp are  much better than R-Z, in general) and better hail identification.  The  latter may more broadly be expressed as better hydrometeor  identification.  
There are a number of polarimetric signatures associated with storms and supercells  that have been identified (e.g. 
Kumjian and Ryzhkov 2008, 
Romine et al. 2008, 
Kumjian et al. 2008) -- Zdr columns, rho_hv and Zdr rings (or  half-rings), Zdr arcs, Zdr and Kdp columns, and the tornado debris  signature (TDS).  As Bart noted, the TDS is only really seen when the  tornado is very near the radar -- we see it quite a bit in our mobile  radar data, but we're also always <30 km from the tornado.  There are  several cases in WSR-88D data (from KOUN), as well as some C-band  radars (the 5/10/10 OU-PRIME dataset has some good examples).  The Zdr  arc is currently thought to be caused by drop size sorting resulting  from rain falling through an environment with non-negligible  storm-relative helicity (
Kumjian and Ryzhkov 2009). 
What I think many on this board are looking for, myself included, are signatures that  provide some prognostic ability.  Joey Picca presented some preliminary  work at the 2010 SLS in Denver a couple of weeks ago looking at the  volume of the Zdr column relative to hail and tornado occurrence (see 
Picca and Ryzhkov 2010); Matt Kumjian also showed some results from an examination of the polarimetric characteristics of hail (see 
Kumjian et al. 2010).  My  dissertation research is focused on the time evolution of polarimetric  signatures (essentially examining how the Kdp columns, mid-level rho_hv  rings, etc., are affected by changes in CAPE and shear), though I'm only in the relatively early stages of this.  Some images of nice examples of some of these polarimetric signatures (primarily those that are seen >3 km AGL) are available in our conference paper (extended abstract) from SLS - see 
Snyder et al. 2010
FWIW, I also contend that most people away from the warning desk (heck, perhaps even at the warning desk) will find it quite difficult to make sense of Kdp, Zdr, and rho_hv data in a real-time situation (e.g. while chasing, arm-chair chasing, etc.) .  In isolation, save for perhaps Kdp, it's difficult to know what the data mean.  For example, to get the most value from Zdr, I think most folks will want to see Zdr AND ZH, or rho_HV AND ZH. Again, the rainfall estimates should be considerably improved, and the hydrometeor classification likely will be popular, but I think the individual products will be considerably less popular with most chasers.  When you have the time to thoroughly examine all products, certainly, you can learn a lot more about the hydrometeors / scatterers in and around the storm, however!
EDIT: Freezing level identification may also be useful, particular in winter precipitation situations -- Zdr and Kdp are very useful in this case!