With the possible exception of hockey & baseball, you'll see players flopping from time to time in order to get calls they don't deserve. I couldn't find the video on YouTube, but wasn't it Geno Smith who did what was probably last NFL season's most laughable flop when he pretended he got hit in the head by a DL who jumped over the line prematurely? It might have been more believable if his reaction was so delayed. When I started watching football in the late 70's punters were taught to flop anytime someone came anywhere near them. The NFL put a stop to that by passing a rule that made it a 15 yard unsportsmanlike conduct penalty for trying that.
As far as the 2 sports that I think are in the running for being the "floppiest," that would be basketball & soccer. In terms of quantity, I think its to close to call. In terms of percentage of flops that are so obvious that you don't need replay, I think its basketball hands down. In terms of who goes to the greatest lengths to sell their flops, its soccer hands down. Seriously, I watched the Cote D' Ivoire vs. Japan & the 2nd half of USA vs. Ghana. I swear that there's at least a half a dozen time per game in which players were so dramatic that you'd think that they broke their leg, shin, ankle, etc.; only for them to get up after a few second & run off without any limp what so ever. I've never seen anything like this in any sport I've ever watched.