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I think this is an interesting question, which I'd love to hear the answers for from those who are actually good at fewest moves.

I've always thought it would be nice to have a very structured process for doing fewest moves, but it seems like my efforts to do so have not resulted in very good results.

I unfortunately am pretty bad at it, but here is my typical process:
1. Use the regular scramble. Sometimes I will eventually try the inverse scramble if it's going really badly, but I find that I don't do it very often anymore. The primary reason for this is that I find myself looking for premoves often at the beginning, and those tend to find the same easy solutions that you'd find with the inverse scramble, so it increasingly seems like a waste of time to do the inverse scramble at the beginning.
2. Look for an easy 2x2x2 block. If there are several corner-edge pairs, possibly look for a direct solution of a 2x2x3. But I find that usually doesn't work all that well for me, so I probably shouldn't bother with that so much. When looking for a 2x2x2 block, also look for a pseudo block, in case premoves can still get me there quickly.
3. Hopefully I found a 2x2x2 block, so extend to a 2x2x3 block. Consider all 3 corner extensions and see what's possible. With each of these steps (2-5), I will always keep an eye out for premoves that can help. If I can't find anything good here, I'll typically try NISS. But assuming I find something, go on to step 4.
4. Extend to a 3x cross. It should be possible to find something in a reasonably small number of moves. If I can't, again I'll try a NISS switch.
5. Look for as many ways as possible to solve the F2L. Hopefully there are multiple final insertions possible. For each one, check all possible OLLs that orient the edges to see if any of them leave just 3 corners unsolved. Hopefully I can find one.
6. If I found one, look for a good insertion, and done.

The problem with this method is that it generally averages around 35 moves. It's pretty reliable, but not good enough to really be competitive. Of course I will look at other things if I see them; I love finding simple skeletons, solving edges directly from the 3x cross, etc., but usually that doesn't happen - I'd estimate at least 60% of my solutions fit the pattern above.

So does anyone else want to try giving their detailed approach?

Using NISS, premoves and insertions
Typically block build my way op to F2L-1
Then manipulate LL via the remaining slot to leave 3 to 5 corners

Methods that are very usefull to know for FMC: Petrus and Heise.

Both start out with Blockbuilding, then orient edges.
Petrus: EO after 2x2x3 block
Heise: EO after (or during) F2L- 1 slot
Both ways of EO are good to explore during a solve

For the end game Heise provides you with some good tactics for leaving 3 or 5 (easier to get to) corners.
In a normal Heise solve you then solve these corners via corner 3-cycles (commutators)

For FMC you insert these 3-cycles into earlier parts of the solution.
This way you can typically use 8 move optimal commutators (instead of up to 12 move optimal when you solve them at the end of the solution)
The real neat trick is to insert these cycles in such a way / location that they cancel moves.
For a 3 cycle insertion you can expect to cancel 2 moves: then you have solved the corners with 6 HTM
For two 3 cycle insertions you can expect to cancel about 6 moves: then you have solved the corners with 10 HTM

Using NISS, premoves and insertions
Typically block build my way op to F2L-1
Then manipulate LL via the remaining slot to leave 3 to 5 corners

Methods that are very usefull to know for FMC: Petrus and Heise.

Both start out with Blockbuilding, then orient edges.
Petrus: EO after 2x2x3 block
Heise: EO after (or during) F2L- 1 slot
Both ways of EO are good to explore during a solve

For the end game Heise provides you with some good tactics for leaving 3 or 5 (easier to get to) corners.
In a normal Heise solve you then solve these corners via corner 3-cycles (commutators)

For FMC you insert these 3-cycles into earlier parts of the solution.
This way you can typically use 8 move optimal commutators (instead of up to 12 move optimal when you solve them at the end of the solution)
The real neat trick is to insert these cycles in such a way / location that they cancel moves.
For a 3 cycle insertion you can expect to cancel 2 moves: then you have solved the corners with 6 HTM
For two 3 cycle insertions you can expect to cancel about 6 moves: then you have solved the corners with 10 HTM

i also really like the 2x2x3 approach but while doing that i try to make as many corner edge pairs as possible, then EO mostly while trying to leave 3 - 5 corners