Ask on /r/AskAnthropology as haircuts predate the written word by far.

Hair can be washed with soap.

I'm sorry to jump on this conversation, but I've also had this question brewing.

What about other parts of hygiene like trimming nails? Could we do that during the stone age?

This would be a better question for /r/Anthropology , but I will say that the oldest images we have of people from previous civilizations are cave paintings and none of the ones I have studied seem to depict people with floor length unmanageable hair. From this we can deduce that humanity has been cutting it's hair for at least the last several 10s of thousands of years.

And yes, as far back as the written and pictoral record go, different hair styles have been popular in different regions at different times. Popularity can be influenced by type of hair prevalent in that region (think of how different a black person's hair is from a white person's hair is from an Asian person's hair), what the weather is like there (often shorter in warmer climates, longer in cold places), socio-economic status of the individual, and sometimes just what is going on in history at that time (for example, when Bejamin Franklin was in Paris, the women used their hair to copy the size and shape of his then-iconic fur hat).

Did it correlate with the discovery of fire ?