Personally, my favorite is The Bar-B-Q Shop on Madison in Midtown, which has something no other BBQ joint in Memphis has, to my knowledge: the endorsement of a Nobel Prize laureate. Seriously. In an episode of his "Theme Time Radio Hour" back about 10 years ago or so, Nobel Literature Laureate Bob Dylan recommended that fine establishment to his listeners.
Other noteworthy establishments: Central BBQ (4 locations that I'm aware of), Paynes (or is it "Payne's"?) on Lamar (lauded by some Food Network celebrity "chef" (I forget who) as "the best thing she's ever eaten"), Jim Neely's Interstate Bar-B-Q on Third Street, Cozy Corner on North Parkway, and The Germantown Commissary on S. Germantown Road.
But I'm sure I'm missing a number of worthy choices, which will probably be mentioned in future postings here. In my experience, if you ask five random Memphians which place is the best BBQ joint in the city, you'll probably get five different answers.
Cheers,
bcg