In order to answer your question, I want to first make sure that we understand what a surfactant does, and how it works. The typical goal of a surfactant is to reduce the surface tension of two immiscible (unable to mix) liquids to make them miscible. When two liquids are immiscible, it is typically because one is polar (like water) and one is non-polar (like most organic solvents and oils.) A surfactant is a molecule that has either an ionic or polar end called the head, which is soluble in polar solvents (hydrophilic) or a non-polar end called the tail, which is soluble in the non-polar solvent (hydrophobic.) The surfactant moves to the solvent-solvent interface and orients itself so the head is in the polar solvent and tail is in the non-polar solvent. This reduces the surface tension between the two solvents. A surfactant can also be used at a liquid solid interface in order to change the wetting properties of the liquid, at which point it is sometimes called a wetting agent. However the mechanism of action is the same. In this case the solid surface might be hydrophobic and the liquid polar or vice versa. The surfactant acts in the same way as before, reducing the surface tension, and allowing the liquid to spread on the surface.

So, a surfactant is designed to dissolve in water (or other polar solvents) as well as in oils (or other non-polar solvents) although I am not sure how great the solubility is. From my understanding though, this is a requirement. Before they are dissolved they can be either liquid or solid, but once dissolved, they do the same things.

Hopefully that answers your question, but if I misinterpreted it or wasnt clear please let me know!

Surfactants, by definition are molecules that have one part of them that "likes" water and another part of them that "likes" oily things (there is a big range of materials and liquids in between so this can get complicated). If you make the oily side very long and the part that like water very small it might not dissolve in water. The other way you could do it would be to attach molecules to a surface that you know will not dissolve.

As for surface tension: The reason things have big or small surface tension has to do with energy, the lower the energy of the system the "happier" it is. If a material or liquid would rather be surrounded by itself rather than another liquid/gas/ or surfaces the higher the surface tension. Water has a high surface tension because it has a very favorable interaction with itself it likes to be around other water molecules rather than be around air. When you add the surfactant the oily tails would much rather be next to the air than the water. Also, the end of the surfactant that "likes" water would rather be next to the water than the air. This causes the surfactant the stick its oily tail up toward the air and its water liking head down to the water. This lowers the energy of interaction between the air and the water, which lowers the surface tension.

There are oily surfactants that sit at an air-water interface. Long-chain fatty acids are solid surfactants that don't don't dissolve in water and sit at an air-water interface.

There are also long-chain fatty acids with alternating double and single bonds that can sit at an oil-water interface. When they are polymerized with ultraviolet light, they form a solid film on the water.