Insecticidal soap is sold commercially for aphid and other small insect pest control; These products may not always use the word soap, but they will list "potassium salts of fatty acids" or "potassium laurate" as the active ingredient.
Certain types of household soaps (not synthetic detergents) are also suitable for pest control but it may be difficult to tell the composition and water content from the label.
Potassium-based soaps can be soft or liquid.
For years I recommended horticultural oil and insecticidal soap for sprays, and thought they worked equally well against all species. A recent article by Cliff Sadof of Purdue University and his graduate student, Carlos Quesada, in HortTechnology(October 2017, volume 27, page 618-624) shows me that I need to update my recommendation.
Carlos and Cliff did a series of lab and field studies on two armored scales (pine needle scale and oleander scale) and two soft scales (calico scale and striped pine scale). Oil and soap, both applied one time at 2%, killed 67-93% of crawlers of all four species; that’s a pretty good level of control. But both oil and soap became less effective as the scale insects settled comfortably and grew. Spraying oil or soap against adult scales was as good as spraying water. No surprises so far. The basic recommendation still applies: You need to spray against crawlers to achieve the best control.
Here’s the good part: In the field studies, oil was more effective against settled armored scales, whereas soap was more effective against settled soft scales. Who knew there are differences between oil and soap on which group of scale insects they are most effective against? I didn’t!
Carlos and Cliff speculated that the difference arises from the chemical properties of the chemicals and the scale insects. Both oil and soap kill mainly by suffocation, but, chemically speaking, soap is polar (so it likes to stick to another polar object) and oil is non-polar (it is repelled by a polar object). As armored scale crawlers settle, they produce a waxy cover over their bodies within three days. Most soft scales, on the other hand, do not usually produce a thick wax layer until adulthood. Wax, being non-polar, reduces penetration of polar soap but allows penetration of non-polar oil. Skin of soft scales is polar, so soap sticks and penetrates the layer more effectively, thus doing a better job of killing soft scales.