Exactly as it sounds- be it ingame or on shoddy, whether hacking or just putting the finishing touches on something you trained yourself, do you like to have the four moves ordered in a specific fashion?

I tend to do that. Status -> Stat boosts -> STAB attacks -> all other offensive moves -> Recovery.

In any game where the poke in the #1 slot is not necessarily the lead, I usually order them based on their type, though I guess kind of subconsciously BST plays a part in it too, since Metagross somehow always winds up as #6.

Since I haven't really explored online yet, my only experience is in-game. There, I order my moves according to usage. I put the most useful move in the first slot, and work from there. I do so because I don't use the stylus for play, I use the buttons. It is faster to just put the most useful move in the first slot and tap A twice at the start of a battle.

In-game i do. I usually put the longer moves so they're on bottom and small moves on top sorta like this:
Pikachu (as an example)
-Hyper Beam
-Superpower
-Thunderbolt
-DynamicPunch

And yes, that moveset is a joke.

Eh, not really. I tend to order my moves haphazardly. When I make a team, however, I always tend to place my pokemon in a certain order when getting them out on Wi-Fi, although I'm not entirely sure why.

I order my moves so that attacking moves are on the left/top, and support moves stay near the bottom/right. My pokemon are always ordered by their roles. Starting with sweepers, and getting bulkier and bulkier as we go down the list.

No, not really.

If I'm just using a set yanked from here or Smogon or whatever, the usually end up in alphabetical order though.

Usually: STAB Moves, No STAB moves, Status and support

for walls and support : status moves and such come first then everything else

Garchomp @ Salac Berry
Jolly - 252 Atk/252 Spd/4 Def
- Substitute
- Swords Dance
- Outrage
- Earthquake

I put them in the order I tend to use them in. :D

I tend to do:

-Stat Boost/Drop move
-STAB/Primary attack
-Secondary attack / status
-Tertiary attack / status / recovery / Other

Example:

Steelix @ Leftovers

-Screech
-Gyro Ball
-Earthquake
-Stealth Rock

(set works much better than you'd expect, btw)

My team tends to be organized like this:

-Lead
- Phys Sweeper (if not the Lead)
- Spec/Mixed Sweeper (if not Lead)
- Phys Wall
-Spec Wall
-Support / Other (if not lead)