Which of these is the safest: AMP, Indy, or Agency?  

Probably the safest of all would be a well-reviewed indy who screens carefully, keeps a low profile, and habitually deletes emails with a security program and does not keep written records.  I just doubt that such a person exists, as some of the recent questions on this board attest.

AMPs are appealing because they are so anonymous. The client never has to worry about calls to home or office. If they ever get busted (and most do sooner or later!) there's no little black book or computer file to get confiscated.  If a client was actually there during a raid, I would guess that LE would pretty much have to catch you in the act, wouldn't they? Most AMP providers lock the door to the session room, giving time, I would think, to don some clothes.  

My guess is that agencies are the highest risk of all. They make attractive targets and probably keep the most records.

These thoughts are just hunches.  What are the opinions of the members here who actually know something!?