As 2021 (soon 2022), what are the regulations for moving to the UK for work?
For most EU citizens, this will not be possible without a sponsoring employer, which is excluded from consideration. Another possibility that seems to be beyond the scope of the question is family-based immigration.
Is registering as self-employed acceptable?
Registering as self-employed doesn't help one to qualify for a visa.
Would creating an ltd and working through it make things easier or would this be bending the rules?
Setting up a new business can help one qualify for a visa, but the business must be "innovative" and must be endorsed by a sponsoring organization.
Can they move there and just start searching for work? (staying less than 90 days in a 180 days period).
There is no 90/180 rule in British immigration law. Visitors are typically admitted for six months at a time. It is permissible to look for work as a visitor, but immigration officers seem not to know this, and there have been reports of EU citizens being denied entry to the UK under these circumstances. Regardless, anyone who does find work while in the UK as a visitor is prohibited from working in the UK. Visitors are also prohibited from switching to another visa category, so it is necessary to leave the UK, apply for a work visa, and wait for the visa to be granted before returning to the UK and starting the job.
The UK has an online visa check tool that listed five possibilities when I used the example of an Austrian citizen wanting to start a business:
start-up or innovator visa: for one of these visas you need an "innovative" business idea that "must be something that’s different from anything else on the market," which you will need to demonstrate by getting an endorsement from an approved endorsing body.
global talent visa: to qualify, you must be a "leader or potential leader" in academia or research, in arts or culture, or in digital technology.
representative of an overseas business visa "if you work for an overseas company and you’re planning to set up your first UK branch or subsidiary."
investor (tier I) visa "if you want to invest £2 million or more in the UK."