Surfactants purified by recrystallization. Sodium dodecyl sulfate has impurities of dodecyl alcohol and sodium sulfate. All components are dissolved in hot ethanol, and while cooling the crystals appear only SDS. SDS crystals were separated by filtration. So repeated several times. It can be purified by dialysis, column chromatography, etc.
The purity of the SDS is checked by the absence of a minimum on the isotherm surface tension / concentration and the peak of light scattering/concentration around the CMC. Small presence dodecyl alcohol causing these undesirable effects.
You can combine analysis and purification based on a procedure of Lunkenheimer.
It is based partially on effect mentioned by Yuri, minimum in surface tension at cmc. It concerns purification of impurities with higher surface activity.
When you enlarge interfacial area volume ratio there will be relatively fast adsorption of the impurities. Then a step of compression of the interface will follow with surfactant molecules to be purified will desorb. This could be done in Langmuir trough or turning a specially designed container named flounder. You would the suck off the interfacial layer and start the whole process again until surface tension stays nearly constant.