Fastest way to unlock custom special moves?
Classic mode is a good way to get special moves, and seems to be the fastest for the 3DS version of the game. You have a slight chance to get custom moves for any unlocked character (except Mii and Palutena, who start will all moves unlocked, and DLC characters, who don't have any custom moves), a higher chance to get custom moves for the characters you fight against, and the highest chance to get special moves for the character that you're playing as. Keep that in mind when selecting your character and each battle you pick. You can generally get more on higher difficulties, but you'll lose some rewards if you lose both lives on one stage. You can also use custom characters to make completing Classic mode faster and easier. On the 3DS, you can use a home button to temporarily freeze the spinner before each match, which makes getting custom upgrades, such as special moves, from the spinner easier.
All-Star mode on the Wii U version with the difficulty set to Hard is a great way to get moves for the specific character you're playing as, because completion of each round gives two rewards, special moves are often the rewards, and dying will not lose any rewards you've already earned. Unfortunately the 3DS version provides fewer rewards, and they are mostly trophies, so this is not a good way to earn special moves on the 3DS.
Another way to get special moves is to have an Amiibo fight computer players. Amiibos gather upgrades, including custom moves, when they fight. You can use this strategy to get new moves while away from the game.
On the Wii U version, you can also get special moves from Special Orders. Each special order you select gives a hint at what you'll receive for completing it. For a higher chance to get custom moves, select tickets that show something other than money or the Super Smash Bros icon.
Also on the Wii U version, you can do Smash Tour to get larger quantities of rewards, including special moves. The downsides to this are that it takes longer than the other methods to complete a single round, and that you have less control over the resulting rewards.
I only have the Wii U version. I've collected 97% of the collectable custom moves and 95% of the total custom moves. This is what I suspect to be the most efficient way to collect custom moves on the Wii U.
General info:
Classic mode seems to favor giving R.O.B, Sonic, Donkey Kong, Diddy Kong, and Falco custom moves.
Although people seem to enjoy bashing the trophy rush game, it is probably the most time efficient. I would suggest waiting until you've unlocked about 80% of the custom moves and acquired a massive amount of money because it is by far the least money efficient. Make sure that you use whatever character that you're trying to get custom moves for.
If you are bad with a character, wait until you've collected about 80% of the custom moves and acquired some money. Then you will want to use the character that you are bad with in trophy rush, master orders, and crazy orders.
In classic mode, the order of priority for what you are trying to get on the spinner should be as follows: Custom part, Crazy orders pass, and you can choose whether coins or trophies are more important to you.
There are some custom moves that can only be unlocked by completing a challenge, use this website to check what challenges you need to complete in order to unlock custom moves: http://www.ssbwiki.com/List_of_challenges_(SSB4-Wii_U)
Step one: The first thing that you will want to do is create a table so that you can keep track of every character's custom moves.
Step two: For the first 40% of unlockable custom moves I would suggest using mainly smash tour. For every two or three rounds of smash tour I would suggest playing one or two rounds of Classic mode (difficulty at least 3.5) or all-star mode (normal difficulty).
Step three: Once you've collected 40% of the custom moves, I would suggest playing classic and all-star mode on the highest difficulty that you can manage with the characters that you're trying to unlock custom moves for. You can also start playing master orders and crazy orders with the characters that you're trying to unlock custom moves for.
Step four/custom moves for characters that you're bad with: Once you've collected 80% of the custom moves, you will want to start using crazy orders, master orders, and trophy rush almost exclusively. You should be able to afford it if you've been saving up money and crazy order passes. If you ever feel like you're running low on money or passes, switch to classic and all-star for a while and focus on coins and passes.