I figured I'd start a thread for people to work together to compile a list of Badge Arcade boards that benefit from the LR trick - primarily the ones that benefit directly from it from the start, but also the ones that can allow you to get all badges using just one turn to loosen the non-loose badges.
For those who don't know the LR trick, watch this. For what follows, I will mark it as LR if you benefit from starting by going left from the board (that is, moving from the board to whichever board is to the left of it, and then back again, repeatedly), and RL if you start by going right.
Remember, for all of these, you can perform the trick without having any turns, but the pits only open when there are turns available. So feel free to try it out without any credits, but you'll need to either win some free turns, or pay for some turns, to collect your rewards.
I'm Australian, so my contributions come from the European version's lineup (don't expect me to post up stuff that hasn't been released in EU). I encourage others who find cases where the technique works to post theirs up, too. To get the image, you can go to the board, then switch to miiverse and post in your screenshot album, then access it via your computer and transfer to an image server of your choice (I've used photobucket, except for the thwomp-containing Mario Kart one, which I found online)
Animal Crossing: Mabel, two animal villagers, two dresses, with an arrow as part of the board
Using LR (as seen in video linked above), you can slowly edge the green dress and two animal villagers to the right. As they move to the right (the green/yellow obstacle should shift right along with them), they'll start to push against the red dress, which will then start edging the red/yellow arrow obstacle out. After some time, the right animal villager will fall onto the arrow as well, and with a little more patience, the red arrow will loosen and fall into the pit, along with the red dress and both animal villagers. Mabel and the green dress may not drop immediately as seen in the video, but may instead get caught with the green/yellow obstacle stuck slightly. To loosen it, keep trying RL and LR flips until it drops. You can COMPLETE this board without ever using a turn, so long as you have a turn available at the time (if not, the badges will just get stuck at the bottom of the screen).
Classic Mario underground + hammer: with Fire Mario, Blocks, Hit-Blocks, ? Blocks, Fire Flower, Buzzy Beetle, Buzzy-hit, and fireball.
[Note: image already has trick used] Using LR, you can slowly shift the top ? block to the right, until it drops onto the next ? block down. Keep using LR and it will edge to the right. When it topples off, it should hit the Buzzy Beetle and two blocks in the bottom right, to score you four badges. You may also inadvertently have the fire flower fall to the left, which may either help or hinder if you want to try to clear it.
Mario Kart: with Thwomps, Shy Guy, Koopa Troopa, Lakitu (racer), banana, green shell, and boomerang flower.
Using LR, you should be able to slowly nudge the rightmost Thwomp over the edge - when he goes, he should take Lakitu and the boomerang flower with him. I wasn't able to get the other Thwomp to go over with continued effort, though.
Mario Kart: with Peach/Daisy/Rosalina on bikes, Toadette in kart, two ? item blocks, spiky blue shell, and super horn.
This one's a special case. Upon entry, the ? block that is (in this image) on top will actually be below the crane, and will immediately drop onto the bouncy red obstacle. If you let it move, it'll get stuck where you see it in the image. However, if you LR or RL once immediately just as it hits the bouncy obstacle, it'll lose its momentum, and when you return, it'll fall straight down into the pit.
Requiring one turn to loosen badges:
Any board with a single slope straight into the pit (examples here, Mario Kart Koopa Kids, and Mario Kart Babies + Cow and Cone):
For these boards, you'll need to use a turn to dislodge the items. Once they've been dislodged, however, using LR (in the cases shown - there may be ones with the slope going the other way) will slowly inch the badges down the slope. You should be able to get most of the badges in the case seen above (possibly all of them, but it'll take time if it can be done) - I gave up on the Koopa Kids after I got the first four, and on the babies before I got Baby Mario.
Animal Crossing: Streetpass Plaza, Music, eShop, and Mii Maker action badges.
For this one, they're all non-loose, so you'll need to loosen them. If you take the eshop badge (with Tom Nook), with proper placement, you'll also knock the Mii Maker badge (with Reece), making the platform tilt to the left (you can probably do this in reverse, taking KK), while also bumping KK, who should dislodge Digby, but they probably won't drop into the pit. However, using RL will edge them down.