I've encountered many, many mobile games where the character needs to perform five different actions, so the developer adds five separate buttons to the screen. Of course you will mis-tap them and die in the middle of a boss fight.
The best touch controls are achieved when the dev designs the game around touchscreen, not attempts to adapt touch controls for some existing game.
For platformers there are two movement buttons on the left, and three buttons on the right part of the screen - jump, attack, and alternate attack or some action like dodge. Any more buttons make the game hard to play. There is also a common mistake of making buttons the size of a thumbtack. Ideally the buttons should be as big as a 5 Euro coin, that would be a third or even a half of screen width for most phones.
My recommendations are SuperTux and Swordigo.
For twin-stick ahooters there are two joysticks, and maybe one or two extra action buttons above the right joystick, but not anything more.
The best examples are Space Marshals and Crimsonland.
Top-view RPGs and dungeon crawlers also tend to use twin stick controls. The gameplay tends to be more relaxed, because you can slways grind few more levels and don't bother dodging enemy attacks.
Shoot-em-up is another type of game that works really well with the touchscreen. Your aircraft follows your finger no matter where you touch the screen, it's simpe and it works well. There is a wide variety of quality shmups on Play Store, try OpenTyrian for some classic DOS gameplay.
Honorable mention to swipe controls. You can swipe up/down/left/right without aiming for a specific button and even without looking at the screen, ao it's impossible to mis-tap the wrong button. The downside is that swiping is slower than taps, so the gameplay tends to be slower. Reaper is a good example.
First person shooters are okay for casual gaming, but playing any competitive Counter Strike clone like Critical Strike or Critical Force will earn you a friction burn on your finger, because you are swiping the screen non-stop to aim.
I'm not reviewing strategy games here, they can have 10-layer menus and dialogs and still be playable.
Some racing games support gyroscope as a replacement for the steering wheel, it works rather well.
And of course there are infinite runner games. I don't want to call the whole infinite runner category trash, there are some good runner games like SmashHit or Vektor or Alto's Odyssey, but if it's three lanes infinite runner, you will watch ads each 30 seconds, and the gameplay is only fun for the first 30 seconds.
Flappy bird. Best touch controls ever, but the game itself is garbage.
There is a specific class of mobile gamers who are using gamepads. The gamepad is great for sure, you have a separate button for each finger, however the gamepad is more often than not bigger than the phone, so you are losing convenience and need clothing with huge pockets.
Leap Day by Nitrome. Tapping anywhere on the screen does exactly one thing, jumping. It's a vertical platformer. You run back and forth automatically, turning around whenever you hit an obstacle. 15 stages to a level and there's a new level every day. You can play any of their backlog of levels going back a few years as well. There's some pretty imaginative level design, but if you play for several weeks you start to see a bit of repetition. It normally takes me between 20 and 40 minutes to get through a level.
Another good one is Rider, and its sequel Rider Worlds. Side scroller car obstacle course with a neon/Tron aesthetic. When you're in contact with the ground touching the screen makes you go forward and accelerate. When you're not in contact with the ground, touching the screen makes you rotate. Score points for flipping in the air and collecting gems. There are a series of increasingly difficult challenges as well.