i don't gig much , for a few years, now; but when i did it was flatbased all the way and will be in the future, when i pick it up again.
the generalized idea ,that flatbased stands are flimsey and bound to break , so you need stands that are built like a tank, is a myth. what is the relationship between the leg format and strength or stability? there are tripod stands that are cheap crap and there are flat based stands that are cheap crap too. as far as stability is concerned, that is primarily a factor of leg length and the overall design of the clamps and upright tubes. tripod stands can have short or flimsey legs , light duty clamps and light guage tubing, as can flat based------so from the stand(DOH)point of strength and stability, they are potentially equal.
where flat based stands win hands down is in compactness of settup , ease of settup and breakdown and compactness when transported and stored. the problem with flatbased stands is ,---- that most of the really good ones are no longer being made. they are out there and often quite cheap, though. most are of European mfg., so shopping for them on the Wichita craigslist, might be futile. if you want really great ,light ,strong flatbased stands , look to Europe . i just bought 12--a couple of hi-hats included, from 3 different sellers out of Germany for a grand total of around 300.00 including shipping----some of them are almost new. they will go with some very nice vintage kits for a whole new generation of vintage enthusiasts to appreciate.