I know that in Italy some people were using a solution involving wood ash, but from what I understand this is only known to have been used in that part of the world. People around the world learned how to extract oil and animal fat and I know in some societies where access to meat was difficult, oils were among the main sources of protein for the poor. There's also the industrial use of needing to de-grease work surfaces in various industries. Like what did a blacksmith or an olive press worker do to clean off at the end of the day?
I have seen a couple sources on the history of soap and other cleansing agents, but right now I can only find one: Chapter 9 of Chemical Technology in Antiquity, "An Ancient Cleanser: Soap Production and Use in Antiquity" by Kristine L. Konkol and Seth C. Rasmussen, 2015.
They note that the simplest cleaning agents are just abrasives like sand and ash, but that there are other "soapy" analogs which were used before soap was widely manufactured, particularly focusing on saltworts and soapwort.
Soapwort usually refers to the common soapwort, Saponaria officinalis, which is full of glycosides called saponins which form a foamy soap-like froth when dissolved in water.
Saltworts are any of a number of plants, mostly halophytic costal and desert plants in the Chenopodiaceae family, which accumulate alkaline salts in their tissues and can be burned to produce either potash or soda ash, depending on the plant. Two saltworts mentioned in this paper are Salsola kali and Salicornia herbacea, but there are many others as well. Both potash and soda ash have been used as cleansers and as ingredients in cleansing solutions.
Potash (potassium carbonate) could also be derived by leaching wood ashes of some trees.
Soda ash (sodium carbonate) is still used in a lot of products today, especially dry powder soaps.
Another detergent compound which was extensively used in Egypt (as well as being exported throughout the Mediterranean) was natron, a mineral evaporate (sodium sesquicarbonate) collected mainly from the dry lake-bed of Wadi Natrun. Natron was used for cleaning as a sort of proto-soap, but also as a cooking salt, a preservative, medicine, a desiccant in embalming, a chemical agent in ceramics manufacture, and as a tooth-cleanser and mouthwash. It also produces smokeless fuel when added to castor oil!
Konkol and Rasmussen mention that the Sumerians had no specific word for soap (understandably, as it seems proper soap hadn't been invented yet) but that the same cuneiform symbol is used "for the calamus plant (a type of soap plant), for potash, and for the idea of washing" citing M. Levey, "The Oldest Soap in History," Soap Chem. Spec 1957.
I can't find that paper, but I do want to linger on that for a minute. The sign they must be referring to is "naĝa", [𒉀] which the EPSD gives as "Potash, soap (Akkadian uhulu), with related terms "naĝa dub", "to rub"; and "naga sub", "to rub with soap". The plant name which they identify as calamus must be "naĝa-si-e3" or "naĝa-si" [𒉀𒋛], again linked to Akkadian uhulu, "an alkali-rich plant, potash".
Perhaps also relevant are the Sumerian words "dinig", for salt or potash, and "nimur", which could mean alkali, potash, coal, ashes or charcoal.
Anyways, getting back to Konkol and Rasmussen, they say:
Simple detergents that were commonly used in ancient Mesopotamia to cleanse were alkalis, clays, earths, and resins, although the alkalis, such as those leached from the ashes of plants, were most common. A Sumerian tablet found in the Hittite capital of Boghazkoi discusses the use of soda for cleansing the body:
With water I bathed myself. With soda I cleansed myself. With soda from a shiny basin I purified myself. With pure oil from the basin I beautified myself. With the dress of heavenly kingship I clothed myself.
A variety of plants were used for their alkaline substances in Mesopotamia. In an Akkadian text belonging to a private citizen during the reign of Ashurbanipal (7th century BCE), the author discusses using tamarisk, date palm, pine cone and the mastakal plant (which is unidentified) for their detersive properties:
May the tamarisk, whereof the tops grow high, cleanse me; may the date palm, which faces every wind, free me; may the mastakal plant, which fills the earth, clean me; may the pine cone, which is full of seed corns, free me [...] I carry a container with an aqueous solution of mastakal plant to the gods of the heavens. As I would bring forth to you for purification, so will you cleanse me.
Previous I had wondered if mastakal might not be mastic, a resin from the mastic tree Pistacia lentiscus, but it seems that doesn't check out etymologically (mastic coming from the greek for "chewing" rather than from a hypothetical Semitic trilateral). Recently, Barbara Böck identified mastakal somewhat non-specifically as "soapwort", in chapter 7 of The Routledge Companion to Ecstatic Experience in the Ancient World, "Mind-altering plants in Babylonian medical sources," 2021.