I know that scissors were invented as soon as metal working appeared, but there are plenty of societies that did not have metal working and yet they were never depicted by the explorers who contacted them with messy or long hair, I'm thinking of people in the Americas, the Mayans, the Aztecs, the Incas, but also the Polynesians, they had their own standards of beauty and elegance and they developed hairstyles accordingly... and they did that without scissors.
How did this happen?
You can cut and shave hair with a sharp-edged stone tool, particularly a freshly-knapped stone blade. In North America, there are ethnohistoric references to stone blades and knives, combs, plucking, and burning to cut, style or remove hair. Plucking was common for removing head and facial hair, often using clamshells as tweezers. There are reports from many Native American groups, particularly in the West and Pacific Coast, of burning or singeing the hair short as a sign of mourning.
In a discussion of sacrificial hair cutting, MacLeod has one specific reference to a Wappo Indian woman in California cutting her hair with "a flint knife" (1). A more detailed description of a ceremony for two-year-old children comes from an ethnography of the Lipan Apache of New Mexico and Texas (2).
First they paint the whole head with dark red paint...For this ceremony they go to an old man or woman who has been cutting hair in this way...He uses a flint knife to cut the hair. He leaves it on in five places on the boy--a tuft on top, one over each ear, one in front, and one in back. A little bit of hair is left to give good luck.
In the early 1600s, Fray Juan de Torquemada described the use of obsidian blades by the indigenous people of Central Mexico for grooming. Note that he mentions the Spanish using this method as well in the early Colonial period. The linked article shows a drawing from the Codex Mendoza of a young man's hair being cut by an obsidian blade.
They can cut and shave their beard with them [obsidian blades] the first time, and with the first cutting edge, only a little less well than with a steel blade, but at the second cutting they lose their edges, and therefore another is needed, and then another, to finish shaving the beard or the hair. But the truth is that they are cheap, and thus one does not mind using them up. With them, many Spaniards, seculars and regulars, were shaved, especially at the beginning of the peopling of these realms.
(1) Self-Sacrifice in Mortuary and Non-Mortuary ritual in North America, Author(s): William Christie MacLeod, Anthropos, Bd. 33, H. 3./4. (May - Aug., 1938), pp. 349-400
(2) Remuneration to Supernaturals and Man in Apachean Ceremonialism, Author(s): Morris E. Opler, Source: Ethnology, Vol. 7, No. 4 (Oct., 1968), pp. 356-393
(3) Obsidian Polyhedral Cores and Prismatic Blades in the Writing and Art of Ancient Mexico, Author(s): Karl A. Taube, Source: Ancient Mesoamerica, Vol. 2, No. 1 (Spring 1991), pp. 61-70