The following requires TSA, without question AIUI:
1. FAA
PPL or
IR done in the USA
2. FAA PPL or IR checkride done anywhere
3. Training towards the above with an CAA CFI/CFII, done anywhere
but this "depends" because a flight with an FAA CFI/CFII in your own N-reg plane could be towards practically anything else too, or nothing at all, couldn't it?
An FAA
CPL, and some other stuff, is
not subject to TSA. This of course means that any flight that could count towards a CPL is TSA exempt.
Going back to 3. there is no requirement at all for FAA flight training to be done with an
FAA instructor. All ICAO training is allowable for FAA purposes. What you do need is to be
signed off by an FAA instructor that you are ready for the oral+checkride, and understandably no such instructor is likely to sign you off unless he is pretty damn sure you are up to standard, which
in practice means doing the last 3 hours in the last 60 days before the checkride with that FAA instructor, and those 3 hours (or more, as necessary) will obviously be subject to TSA.
This is why the usual scenario is that people get all their training done here, perhaps in a G-reg with a JAA instructor, and then get TSA+Visa and go off to the USA to finish off.
As for the US
Visa, this is unfortunately a grey area all around, not least because of the sub-18hrs/week exemption for a course of study which some sources such as the US Embassy in London washed their hands of when it came to flight training (I have the reference if anybody wants it). In practice you have a great choice: you either try to explain some fine detail of some regulation to a gun-carrying national minimum wage worker of average intelligence at a US Immigration entry point with 50 exhausted holidaymakers queuing behind you or..... you turn up with an M-1 Visa
