In the news recently, a judge suggested that it would be a good idea if Muslim communities in Britain could regulate themselves using Sharia law on areas like family law and some transactions, while not going beyond the sanctions of the laws of the UK.
This and most other such discussions miss the very simple key problem - who determines which jurisdiction applies? For example, suppose a Muslim woman, married to a Muslim man, with children, becomes a Christian. Sharia law, as far as I remember, would claim that the woman was under its jurisdiction as she did not have the authority to convert without her husband's permission. And if she did convert, depending on the interpretation of Sharia, she would either get killed (prevented in the above suggestion) or treated as if dead - divorced and separated from her children with no right to contact them, regardless of whether her husband was abusive (for example) or not.