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?