Spiderman has shown off his speed many times in different stories. He has a tendency to outrun bullets and electrical energy blasts to their targets, which is a testament to his agility. He can also jump extremely fast, demonstrating he can scale buildings in a matter of seconds. In addition to his agility, Spiderman can travel at superhuman speeds on foot when needed. This has been proven when he effortlessly dodges sniper bullets or casually outruns an explosion from its epicentre to a square block away while carrying a man.
However, finding his true top speed on foot has been hard to do as he rarely runs a straight line without being interrupted. Most people cite his ability to run a mile in five seconds, but this isn’t really running. In this case, he was jumping over buildings as he chased down a criminal.
The best example of his speed comes from the upcoming Spider-Man: Velocity, a Playstation 4 game. In the story, he encounters a young woman moving so quickly she might as well be a ghost. This requires him to use his Spider-Man: Velocity suit which gives him the ability to track her down.
The suit can track the speed of her movements, as well as detect any other object that is within its range. It can then predict her location and amplify his reflexes so he can react to the situation quicker than ever. In the past, this has allowed him to outrun an explosion in a metropolis and catch up to a speedster.