No-one seems to have mentioned Rampage yet. While some features didn't quite translate properly (such as Lizzie's spelling and the buses/trains being near-impossible to hit), it was still a classic:
https://www.youtube.com/watch?v=9wDVWMSy398
Thank you very much! I remember a fighting game with characters called Sword, Club etc. but could not remember it's name. (one of the few I couldn't name from back in the day)