Which martial arts prefer the vertical punch? Which martial arts prefer front kicks with the heel?

Im not sure about the vertical punching, but I believe that many Pankration schools teach front/push kicks to connect with the heel.

:)

I think most style have a vertical punch, at the very least korean, chinese, and japanese styles do. As for kicking with the heel, I do remember that human weapon about pankration, but I'm not sure how reliable human weapon is. Also, a front kick with shoes is usually done with the heel, or at least I've been taught to do that since you can't really strike with the ball of the foot.

All styles of karate use both vertical fist punches, and front kicks with the heel.

In shotokan both are only used 'as needed'. Isshin-ryu make unique use of a vertical fist punch, I can't name a style that 'favors' a front kick with the heel. Front kicks usually land with the ball of the foot, or the top of the foot, and perhaps the shins.

Some Burmese styles also use the vertical fist.

Really only tae kwon do and karate are in love with the front ball kick-

most styles front kick with the flat of the foot or the heel.

The thai foot jab lands flat to the stomach and heel first to the thigh.

The closest thing we have to a vertical punch in Itosukai Karate is a Tate Tsuki or upright fist punch.

As to front kick striking with the heel ... if someone is on the ground from a takedown, and they are in front of you, we do apply a heel kick, usually to the sternum. But only because of the angle of the kick.

Kenpo Karate uses it sometimes and certain styles of Silat do but Silat borrows from Wing Chun so I can't really distinguish it lol

Bujinkan Budo Taijutsu uses those techniques a lot and in some of the most effective ways imaginable.