If a Tsum isn't listed here, I either don't have it yet, haven't gotten around to it, or it's not worth the time. Everyone plays Tsum Tsum a little differently, and these are only descriptions of my own opinions and ways I play them. Not everyone will agree with it, so don't take it personally if I hate your favorite Tsum.
Also, check to see if you have a Tsum Problem and see some Imaginary Tsums that should be in the game.
Ariel
Ariel is everyone's favorite ginger. Her skill is like Belle, but a bit weaker because she doesn't have legs so is confined to the water.
Strategy
See Belle.
Baymax
Baymax is really cute and easy to see. His skill is to enlarge a range of bubbles or Tsums, turning them into big ones that are worth more.
Strategy
Your priority is to collect long chains as much as possible to create bubbles. Check your skill level and save up enough bubbles while you are playing until you can activate Baymax's skill. For example, at skill level 2 he does 3, so save up at least 3 bubbles. If there aren't any bubbles on the screen, he just turns random Tsums into big ones that are worth more and that's not so fun.
Beast
Beast is really cool, and not as ugly as he is in my drawing. He attacks the screen in hanger and slashes a large amount of Tsums in a diagonal line. He does take a huge amount of Tsums to activate, so isn't really efficient until he gets at least skill level 2. At lvl 2, he takes 24 Tsums to activate, and with each level he takes less, kind of like Pete.
Strategy
Unlike Olaf, who also collects in a diagonal line, Beast doesn't need gyro to be really good. Play him like any other burst Tsum (see Belle). If you really want to you can use gyro, but tilt the phone opposite of Olaf.
Belle
Belle is my favorite because she gets ridiculous coins with very little effort even at low skill levels. She makes it rain with her dancing, and you don't even have to try that hard to get her to dance...
Strategy
Play with the thought that you only need to collect Belle Tsums. Don't worry about making long chains, just get Tsums out of the way and then spam Belle skill button as soon as she gets close, especially if you just collected a long chain so that the leftover Belles in your chain will charge the next skill meter.
*Updated* HAWAIIAN STITCH
Stitch is a coin collector like Belle, but more fun to play. His skill collects in an arch shape, kind of like he's trying to fly but can't quite make it and comes crashing down. He sucks at skill level 1, and is pretty good at level 2. I think that Hawaiian Stitch will actually surpass Belle in earning coins at higher skill levels because he hits the edges of the screen and you can keep collecting in the center.
Strategy
Turn on Gyro in settings before starting. Right before activating Stitch's skill button, tilt your phone back so that Tsums on your screen fill the top. Be careful not to overdo it or you'll miss a huge chunk on the bottom. While the skill is finishing up, spot the chains in the middle of the screen where Tsums aren't being collected and be ready to collect them before the Tsums fall.
*NEW* Holiday Jack
Remember how horrible Jack was at doing Christmas in the movie? Well, Holiday Jack is just as horrible. Holiday Jack is like as Santa Jack in the movie - he has great intentions, stopping time and trying to collect chains for you, but doesn't realize how sucky he is and insists on doing you the favor. He wouldn't be so bad if he could be leveled up, but unfortunately he comes maxed at 1 level - level suck.
Strategy
Play him as you would any burst Tsum, since you can't do anything during his skill anyway.
Jack Skellington
Jack is from one of my favorite Disney movies, but he is horrible in the game because he has osteoporosis so has to take it easy. His skill is like Donald, except he collects chains instead of single Tsums (a strangely fitting skill), which makes him slightly better than Donald (who is a "Happiness" Tsum, or Sadness Tsum if you want to be accurate). He is useful only for mission 8 in bingo card #6, but that's about it.
Strategy
It's like playing Donald. When you activate his skill put your phone down and use all 4 fingers on both hands to tap as fast as possible... unless you have a small phone or sausage fingers, then just use 3 fingers. If you have both a small phone and sausage fingers, then just use what you can, but it'll suck.
Little Oyster
Who the hell is Little Oyster? You probably don't remember because he was just in the chorus of the Little Mermaid, not even a supporting character. Why is he in the game? To make curious people Google him to see who the hell he is. Little Oyster sucks, but his saving grace is that he's really easy to see for me, so I don't mind when he shows up as part of the group when using a better Tsum. Also, oysters are delicious.
Strategy
Save up coins to buy another premium box, and use that Tsum instead. Unless it's Perry, in which case just use a Happiness Tsum that is actually nice to look at.
*just added* Marie
Marie is the same as Miss Bunny, just white instead of yellow... I think she is easier to see. If you like popping those bubble wrap sheets, you'll like either one of them. She's just OK for coins, but is strangely satisfying to play.
Strategy
See Miss Bunny.
Miss Bunny
Miss Bunny is apparently the wife of Thumper. She's not as memorable or cute as Thumper, but she makes bubbles appear.
Strategy
Collect as much Miss Bunny Tsums as possible and activate her skill... Just make sure you pop the bubbles in a timely manner and don't let them crowd the space or they won't clear as many Tsums.
OLAF
Olaf collects Tsums in a diagonal line because he doesn't have a brain, so isn't capable of moving in horizontal or vertical lines. Actually I kind of like playing him because he is easy to see and collects a decent amount of coins when played right (way more than the more coherent Tsums who collect vertical or horizontal).
Strategy
Turn on Gyro in settings before starting. When the games starts, tilt your screen a just few degrees counterclockwise so that tsums fill the top left corner of the screen. Then, play like you would with any burst Tsum.
Perry
Who the hell is Perry? I didn't know either until I Googled him. He's from some shitty TV cartoon on Disney channel. With so many classic movies to choose from it is a mystery why Perry is in the game. He is, however, pretty easy to see and is so ugly it makes you naturally faster when clearing him off the screen. He basically pecks at the screen taking out random chunks of Tsums, but since he's a platypus and only has a bill (not a beak) and can't fly, he fails miserably and misses.
Strategy
See Little Oyster.
Pete
Pete is awesome. He's the bully who picks on Goofy, which makes me like him more because Goofy sucks ass. Pete punches the screen clearing a huge amount of Tsums from the center, but also takes a pretty large amount to charge his skill up. He gets OK at skill level 2, but I imagine will be really good once his skill level is maxed and can activate more.
Strategy
See Belle.
Sorcerer Mickey
I really love playing Sorcerer Mickey even though he doesn't get any coins. Seriously, he really, really sucks at getting coins. I like playing him anyway because he has awesome music and his skill is fun (you tap the screen to clear surrounding Tsums).
Strategy
Try to clear bubbles as soon as they appear so it doesn't get in the way when you use his skill. When you activate his skill, tap the areas that have more Sorcerer Mickey Tsum so you can charge up his skill again.
Thumper
Thumper is the lovable bunny from Bambi. I've always liked him, just not in the game because his skill isn't great compared to other burst Tsum. He's like a slightly stronger Mickey. Maybe he will get better at higher skill levels.
Strategy
Like any other burst Tsum. See Belle.
ZERO
Zero is the cute pet dog of Jack Skellington who has self esteem issues and disappears when you activate his skill. He's annoying to play and almost as useless when it comes to points and coins, but like Jack, can be used for a few bingo missions (reducing tsums to 3 kinds, any canine tsum, etc.).
Strategy
Collect Zero tsums until skill meter fills up, then STOP collecting Zero tsums and only collect other characters until mostly Zero tsums are filling the screen. Do not activate Zero's skill until the screen fills up with plenty of Zero tsums. It's really hard to resist hitting the skill button when it activates, but this is the only way to make Zero not completely useless.