Superman can fly for 4.89\text{ km}4.89 km4, point, 89, start text, space, k, m, end text in a minute.
How many kilometers can he fly in 3.53.53, point, 5 minutes?
Answers
It's already been established that all of the Flashes can move past light-speed via the Speed Force. As for Superman, he's never shown to move at light-speed while running inside a planetary atmosphere.
This image should answer your question, though. This scan was taken from Flash v2, #220.
Superman and Flash race through the roads of New York; the captions read Flash's thoughts: "My wife can't wait. I know Linda is out there somewhere. And once we look into each other's eyes-- everything will be fine again. Everything will be perfect. Superman's still right behind me. Though his perception hasn't quite shifted up to mine yet. His movements are slightly slower-- but his sheer momentum is incredible. The power in his muscles snaps the laws of physics and gravity in half. In terms of raw speed, Superman and I used to be evenly matched. I'd even say he might've been a tad faster. But that was before I got over my fear of replacing Barry. Before I discovered where my speed actually comes from. And before I learned how to better tap into the Speed Force and take complete control of my abilities. I could steal all of Superman's kinetic energy and stop him cold, but it's be like throwing him out of a car-- one moving at over two thousand miles a second. That combined with Superman's strength, he'd create a path of total destruction from here to Moscow." Superman says out load to Flash "Whatever happened to you. Whatever you did to hide who you are... it's not going to change your relationship with the League." To which Flash replies "It already has."
Flash states that Superman is moving at over 2000 miles/sec, which is also over 120,000 miles/minute, and also 7,200,000 mph. That's Mach 9350. I believe this is that fastest that I've seen Superman move under the Earth's atmosphere. He can, however, break light speed in the vacuum of outer space.
Superman has raced the Flash several times over their history but it is considered a given that if the Flash decided to run at his maximum, Superman who lack the environmentally-protective Speed Force aura, is unable to move at his maximum without causing catastrophic environmental damage due to his moving near light speed.
Superman and Flash race up the side of a building and off the top of it; Superman: "You already sacrificed years of your life for everyone. You've been given a second chance." "It wasn't a get out of jail free card. It was a wish off a monkey's paw." "We don't know that for sure." "You can't stop me." "I've raced you before Barry. I even won some of those races." "Those were for charity, Clark."
Barry Allen reminds Kal-El, even though we have raced several times in the past, ultimately, "I allowed you to participate and ultimately YOU CANNOT beat me in a race. Not now, not ever."
An important caveat to this statement is that Superman CAN move faster than light in space and has been shown to do that on several occasions in the New 52.
A rugguish looking Superman flies through Satrun's rings out in space the caption reads "Eighteen hours earlier." and Superman thinks to himself "I punch through the ice and rocks of Saturn's rings. And I should be elated. After sixty days of tearing non-stop through endless space... I'm nearly home. But the last time I saw Earth, I was fighting Brainiac while possessed by the Doomsday Virus... and the lives of everyone on the planet hung in the balance. But we saved the Earth. I dragged Brainiac into a Black Hole."
Outside of a planetary atmosphere he has proven to be able to achieve speeds faster than light, traveling from Pluto to Earth in a matter of minutes. Whether this is due to him achieving superluminal speeds or by warping space around him has not been discerned or disclosed. His current top speed is estimated by this travel time at 16 to 20 times the speed of light!
As to why Superman cannot move faster than light in atmosphere, I submit to you an XKCD article explaining what happens when something moves at near-relativistic speeds in atmosphere: Relativistic Baseball. Totally worth the read.