As is fairly common in many other European countries, basic Third Party insurance for everyone should be required. Making this mandatory would probably not be effective but it should be 'strongly advised' with gains for those who are covered.
'Tripped on a pavement?' – don't claim off your local council, use your Third Party insurance. Repeat claims can then be tracked, the cost of the premium can go up to reflect this and councils save money. Fraudulent claims would also be discouraged. The range of cover afforded to be reflected in the price of the premium – some 'common' claims to be seen as part of any basic cover. This basic cover should be affordable to all – reductions for pensioners, those receiving certain State benefits to be taken into account. For those receiving reductions, however, more than a set number of claims within a set time-frame would either reduce any 'discount' or attract a higher premium.
A No Claims Bonus, or perhaps rebate, to be an integral part of the cover: no claim in two years equates to 50% reduction for next year or return of 30% of contributions for example.
The scheme could be implemented either by established Insurance companies or, less preferably, by an authorised government agency.
Why is this idea important?
The 'Blame and Claim' culture is not only unnecessarily expensive and open to massive fraudulent misuse, it also removes the sense of personal responsibility from too many people.
In a truly fair society, it should be incumbent on all to take more responsibility for their own lives and actions. Reducing 'dependency' in all its forms and encouraging self-sufficiency go a long way to achieving this.