Just to say, dd is 15 and cut her hair short in year 7, there were only 2 girls with short hair out of 150.
She went shorter in year 8, and now, in year 10, she has a number 2 shaved cut. She did it last year before a big scout trip, and dyed it bright blue as well. I don't think it looks good, but I have never stopped her, apart from checking that she was sure she wanted to do it. She isn't allowed to have it blue for school, so it was cut off again before school started.
It is hair. I would rather she expressed herself through hair, which grows back, can be 100% different in a few months etc, than through piercings and tattoos, which are pemanent. To me clothes and hair are so transitory, and they are a great way for them to express individuality.
Now dd2 who is 12, has alos got ulta short hair. 6 months ago she shaved half, and had that long on one side look. It grew really long and annoying in lockdown, so as soon as the hairdresser opened she had it cut. Shaved on both sides and long/layered on top. She actually looks stunning, she has a long neck and heart shaped face.
Most girls in her year still have long hair. She now looks different, stand out etc etc which is what she wanted.
Their school has no rules abotu shaved heads, if they did they would be expected to follwo them, but apart from that, why would I wnat her to have ahair style that I like instead of one that she likes?