I fish for carp around London in the Thames quite a bit. I like to keep things pretty simple. A half decent medium weight rod and reel, a forked stick stuck in the mud, (sometimes I use two forked sticks to make a pod ) and canned corn. But not just any canned corn. I use the cheapest stuff I can find. It's usually really slimy and smells funny and the kernels are great big things the size of horses teeth. Usually at the dollar stores, three or four cans for a buck. Put two or three kernels on a regular bait-holder type of hook, a split-shot or two, and throw it out in a likely looking spot. Sometimes I'll throw a handful or two of bait out as chum. Set my rod down in my 'pod', and wait for the fun to start.
My absolute favorite way to carp though is wading the shoreline and watching for schools of carp that are actively hunting in the shallows. Then I toss a crayfish out just upstream from them and when they get close give the bait a couple tugs like a crayfish swimming and then watch some huge carp come darting over and slam it, set the hook,big splash, screaming drag,...it's go time. Just like fishing the flats for bonefish....except bigger fish.