How can you mount a regular flash in a softbox?
There are brackets of various types, as well as a number of DIY solutions. Most softboxes have a "speed ring" at the rear of the box, which has a series of receptacles (holes, about 3/8 inch in diameter) into which the four steel or fiberglass rods go, and on the bottom or sides of most speed rings are a hole or two, which is, or can be, threaded, to accept a 1/4 x 20 National Coarse thread on what is known as a "spigot" in the grip/lighting field. Spigots are made of brass, and are available in many types, like 1/4x20 male with 3/8 female on one end, female-female, and so on. Spigots can be threaded onto the tops of tripods or light stands, and also fit down (and upward) into things like Umbrella Swivel Mounts and the bottoms of professional light units like Alien Bee lights or Speedotron and other lights.
These days, the 5/8 inch diameter spigot is pretty standard; years ago, many lights and light stands had the smaller-diameter studs on top and or used smaller-diameter spigots.
Basically, you need a threaded bolt of some kind,and the best kind is called a spigot. That threaded bolt or spigot is threaded into a hole on a speed ring, and that allows the speed ring to be attached to a light stand, a tripod, or to an umbrella swivel mount. A speedlight flash can be stuck in the middle of the speed ring's opening circle, and held there by a racket, or by a ball bungee, or any other way you can figure out how to affix it--rubber bands, tape, styrofoam, whatever.
The Strobist blogspot has a good illustration on using a ball bungee to hold a speedlight in position in a softbox. FlashZebra.com has spigots for sale,as well as umbrella swivel mounts. here is a page where you can see an umbrella swivel mount, a brass spigot, and a ball bungee used to mount a softbox and speedlight
Strobist: DIY Ball-Bungee Speedlight Soft Box Mount