Here’s an interesting list, from Saturday’s Telegraph:
These are the agencies able to run surveillance operations or “covert human intelligence sources”, for example agents, informants and undercover officers – people allowed to authorise are in brackets:
It starts with “Any police force (superintendent/inspector if urgent)” and it ends with “Royal Pharmaceutical Society (fitness to practice dir)”, with (I counted) thirty five items in between those two.
Quite a list.

What might be useful is to work out the number of people entitled to authorise such investigations. (Though the number who could participate/see the results would be much greater.) At a guess, more than 100,000, though it should be possible to do it fairly accurately. It would give the unconcerned public more of a feeling for the loss of privacy involved.