Clarification: I'm not posting here champs with high autoattack damage (Azir soldiers many marksmen, fiora, yi, etc) nor over-time damage (like Teemo's passive). Just the instant damage from abilities. Also I couldn't play every champion since rework, so if someone wants to add something about Graves, Poppy would be great.
Answers following this format: Name (lane): Burst / Movility / Score.
Ahri (mid lane):
Her 4 damage abilities (including a cc with damage amplificator and a true damage skill) make her great for bursting.
Also, with her ultimate she can dash three times to become a very movile champion.
4/5 - Unless the target is squishy or she is really fed, hard to 100%-0% anyone.
Akali (mid/top lane):
To be able to burst with Akali, you have first to hit the Q on the enemy, the plop damage of the passive is pretty high. Also, she has mixed damage (both magical and physical) making a tank build to her more difficult.
For movility, her ultimate dash with a considerable range and her W grants her invisibility in the area where is casted.
4/5 - If she has no target to ulti, or a pink ward is on, she loose movility.
Caitlyn (bot lane carry):
Her remake gives her great damage with the passive. For making the best of this you should go for thunder masterie and the combo should be E + Q + AA (Auto Attack). The E + Q should be both clicked almost at the same time, it's a common Cait combo. With her passive, when you hit enemy with E you gain range to auto at him, so you are at a safe distance and the AA will trigger the masterie since E and Q would stack also and the 3rd stack would have more damage for the passive of Cait and the thunder masterie. And the ultimate, as always, greats lot of damage and massive range.
Okey, so she is not a very movile champion, but at least she has the E to repositionate in a battle and gain some distance with a bruiser.
3/5 - Althought she has burst, and her AA deals nice damage, the movility is not high.
Corki (bot/mid lane):
His Q, ulti and passive gives him nice damage, and he has a lot of mixed damage (physical, magical and true with AA).
With the W he can travel far, and even more with the new package, like from base to inhib or something like that.
5-5 - Nice sustained damage and excelent movility, can't 100%-0% but from distance can leave an entire team bellow half healt in less than 15s.
Fizz (mid/top lane):
With three damage abilities (and with nice damage), well timed combo can burst a not so squishy champ. With well timed I mean, in the moment the shark comes out, with W activated, Q + AA.
One of the most movile champions, with his E you can pick betwwen double jump and juke jump (jump and go back to first place). And his passive ignores unit collision.
5-5 - Even the burst is excelent, the movility is what grant Fizz the 5 score.
Gangplank (top lane):
2 items, one hit anyone. Thats Gangplank with the rework, his barrell just does insane damage. Give him a Trinity and Infinity and he is the scariest champ in game.
For movility, he is pretty fast and can cleanse after eating the orange.
4/5 - Probably highest burst in the game, but low movility compared with other champs in this list.
Kassadin (mid lane):
Stacking ulti he has good damage, full combo including AA with W activated can burst very well.
One of most movile champions, he has a blink every 6s or so.
4/5 - After combo he has to go out or Zhonyas as he is not very fast and has to wait for Cds.
Katarina (mid lane):
She has good damage, her ulti is great, it's kind of tricky because it's not burst damage ultimate as it does damage per second, but anyway she is here, with 4 damage abilities.
Her jump gives great movility and with her passive she can enter and exit fights pretty easily, also gains movement speed if hit enemy champion with W.
4/5 - Her efectivness relies on killing a champion or getting an assist early in the fight, anyway... Movile and good damage.
LeBlanc (mid lane):
Total beast in early game, can 100%-0% with almost any of her combos. For AoE burst Q + W + R. For single target burst Q + R. For CC chain Q + E + R. Always Q first to plop passive.
Her W (and R if used W as last spell) makes her jump and gives the option to go back to the first place. Perfect for juking, chasing, running.
5/5 - Incredibly good damage and movility.
Jump into an enemy and Q him twice (with 4 and 5 stacks). That should kill him or get him low. Terror of ADC's with his ultimate.
Can jump to any enemy from bush or with ulti, which also grants stealth and mov. speed.
4/5 - Can only jump on enemies, excelent burst damage.
Riven (mid/top lane):
A combo with 3rd Q + W + R can melt an entire team down. Also in 1v1 with stacking passive with abilities + AA it's hard to defeat her.
A dash and three mini-jumps, with one of them able to go through some walls.
4/5 - Even with low CD, she doesn't have a high range mobility.
Talon (mid lane):
High base damage and burst with combo. His ulti added to his gapcloser and other abilities can 100%-0% almost every mid laner and ADC.
A jump with decent range and his ulti with stealth and mov. speed make him pretty movile.
4/5 - Lacks on movility when no enemies closer or running away if ulti is down.
Yasuo (mid lane):
His passive gives him a lot of critical chance, so almost every Q should crit. Added to that, the Q + R combo will totally do tons of damage on enemies hit.
Can dash through every enemy unit with almost no CD.
4/5 - Another one that is no movile if no enemies to dash.
Zed (mid lane):
Very high burst, a combo and ulti plop could get almost every champion.
Very high movility using shadows to dance arround.
5/5 - Excelent burst and excelent movility to engage or disengage.