On the theoretical side, probably the most important work would be the development of computer models that show how the rings originated. Is our simple model of the accretion of a gas rich dwarf or material from a passing galaxy viable? Can the differences between narrow rings and annuli be understood in terms of the impact parameters or the type of material that is accreted? Are polar rings stable or transient phenomena? How are polar rings related to dust-lane ellipticals and other similar objects? It should now be possible to directly address these questions using the latest generation of computer modeling techniques.
On the observational side, one of the most important observations would be improved measurements to determine the shape of the dark halos with more precision. Equally important would be observations that allow us to age date the formation of the ring (see WMS for some examples of how this might be done), with the goal of placing the various polar-rings in the proper chronological sequence in order to see how these systems evolve.
I wish to thank my co-authors in the development of the PRC; Ray Lucas, Douglas McElroy, Tom Steiman-Cameron, Penny Sackett and Rob Olling. Ray Lucas also helped determine some of the values in Table 1. I also wish to thank Dana Berry for helping to put together Figure 1.