I recently visited two barbeque places that have been mentioned here. They're a few blocks apart: Leo's (2619 Crenshaw, 305 or 210 bus along Crenshaw, or the 37 bus along Adams) and Tasty Q (2959 Crenshaw Blvd, also the 305 or 210; I don't have phone numbers handy for either). For scientific purposes, at each place I ordered pork ribs with hot sauce and peach cobbler.
Bottom line: Leo's.
The barbeque at Leo's is fabulous. The meat is perfectly, falling-off-the-bone cooked, to the point where it takes a little care to avoid it falling into your lap if you hold it at the wrong angle. The sauce is excellent as well, rich and well-rounded, and just hot enough to leave a pleasant burn on your lips. (They warned me that the hot sauce was "really hot", but no experienced hound would find it so.) I'd be interested to do a blind test between Leo's and Woody's, but in my memory it'd be Leo's that won.
Leo's does have drawbacks. It's a take-out-only stand, friendly enough but basically impersonal. The ribs were a little on the skinny side, and the peach cobbler was nothing to write home about. The block it's on isn't dangerous or blighted, but it's ugly, crowded, and noisy, so if you're hoping to eat your barbeque right away -- and in my opinion barbeque really does need to be eaten right away -- then you're out of luck.
At Tasty Q, on the other hand, context is everything. They have indoor seating to start with. One of the reasons we were put here on earth, as you well know, is to sit down to a steaming plate of ribs as Aretha Franklin declaims from a jukebox, and you can do that here. The Tasty Q people have a big sign outside announcing their roots in Shreveport, and just sitting down to their food is enough to make you wish you were in Shreveport and not in relentless Los Angeles. Other aspects of the context are excellent as well: the peach cobbler has a very nice caramelized edge that knocks the socks off the competition, and I also ended up with a cup of baked beans that were so overwhelmingly intense that one spoonful's going to be enough to hold you until Tuesday.
The problem at Tasty Q, I'm sad to say, is the barbeque. Although the ribs were abundant, I found them a little dry. As to the sauce, I expect that opinions will be divided. The sauce is very dark, like mole. It's not hot, but it's ferociously rich. It starts out good, but it's quite an assault on the system. As Jimmy Dale Gilmore once said, I'm glad to see you coming and I'm glad to see you go. My decision to order a full slab was clearly a mistake from the get-go, but I'm sad to say that I wasn't sorry when I ended up tossing some of it. But as I say, your mileage may vary.
Okay, that was my experience at Leo's and Tasty Q. Next up, hopefully: J&J and Rib Hut.