I always put at least 2 #5 continuous in the bottom of a strip footing. If the footing depth is 18" or greater, I use top and bottom bars with stirrups spaced at 16" o.c. in my opinion, plain concrete does not mean "unreinforced". but there are no requirements for minimum reinforcing in plain concrete.
for a strip footing supporting a load bearing wall, if calculations show that no reinforcing is required for shear or flexure, I use ACI 318, 14.3 as a guide for minimum reinforcing.
based on my own experience, most building officials expect to see reinforcing in concrete and many city/county/state codes required at least 2 #5 bars in a footing, or #5 spaced at 12" o.c. each way. Contractors typically call a strip footing a "grade beam" and they are used to seeing reinforcing in a grade beam. providing some type of reinforcing will certainly eliminate the arguments with a contractor or city/county/state inspector.