Easy method for clearing mission mode.

[JustOneStingray]

If you rush at an opponent with Tomura and do nothing but spam square and keep walking in their direction if they are on the ground you will keep them in an infinite combo until they die or you get hit by a sidekick assist. This also works with Muscular and Kirishima. 

[SKiL_Clash]

Either I'm awful or this does not work, I rush and spam square and it's just your basic attack. 

[PSNP is crap i left]

If you have Endeavor he's a godsend in mission mode. Otherwise, use All Might if you know how to use him.

[SKiL_Clash]

I've been spamming Deku. Seems to work fine, almost all his attacks are close range. So I can just murder them all.

[tmlfan73]

I've been having the most luck with Todoroki and Dabi. Todoroki's quirk special 1 on the ground is great for setups as the ice stays for a few seconds. Most of the time you can use it when your opponent is down and when they get back up and their invincibility wears off, they get frozen by the ice. Of course that won't work all the time as sometimes they'll jump away, but it is a pretty safe strategy.

 

Dabi is good for when you have a condition when you can't guard as he has great range attacks.

 

Also if you are having trouble with the Nomus in map 4, Shigaraki is very useful as his quirk special 3 (forward circle) automatically kills after 4 hits. Since Nomus don't have unblockable or grab attacks, you can just guard until they try and attack, then use the special once their attack animation is done. Then use quirk special 2 (neutral circle) when they are down and they may get caught in the move when they get up, allowing you to hit the forward circle again. If not, just go back to blocking. With this strategy you can knock out that first superpowered Nomu in about 20 seconds without taking any damage, and netting yourself an easy S on that particular mission.