Well single engine air taxi work is permitted but it is limited to day VFR only, which in the UK is rather limiting, hence why no one does it.
To operate IFR and by night a twin is required for public transport ops, further to this 700 hours total time with at least 40 hours p1 multi IFR are needed for insurance (some concessions on this) but more often 1000 hours with 200 multi are often requested.
The single engine turbo prop manufacturers have been protesting for years to the single engine IFR public transport ban lifted, but there is a world of difference in the capability of a SET over an SEP.
A private owner may employ you to fly his SEP aircraft as long as you do not conduct public transport operations and this does open up a very grey area between what is a private operation and illegal AOC work.