I have heard from many people that the Flash is faster than Superman.
But how do we know this? Have we ever seen them head-to-head?
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.
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.
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.
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.
False, Superman can move faster than lightspeed. On Earth he won't fly at lightspeed because this could be catastrophic to the planet Earth, Flash doesn't have that problem because part of his powers prevent any damage to the planet. In space Superman does not have that limitation and can let it rip. In a recent issue of Red Hood and The Outlaws, Superman flew from Pluto, back to Earth in the same amount of time, or faster, than the Outlaws were able to teleport back to Earth. That is 4.67 billion miles, in seconds.
The thing here is that people are concentrating on the running... yes... Flash can outrun Superman because Superman does not need to run, therefore he is not used to it. On the other hand... if he was flying, it might be a different story.
Earth-Pluto (4.67 billion km of distance) in minutes is in the bases of the 17 billion km/h that Batman refers in the JLs (aprox.16xlightspeed). Maybe approx. 20 billion km/h.
In The Flash: The Human Race (1998), the Cosmic Gamblers get the Flash to run in a race for the fastest being in existence, where losing means the destruction of planet Earth. The Flash counter bets with the Gamblers that he can run faster than they can teleport. The Flash then gathers more Speed Force than ever and makes a move in less than a Plank Instant (smallest unit of time), arriving at destination before instant transportation of the Gamblers is achieved.
Being faster than teleportation is certainly faster than Superman, as we've never seen Superman able to catch up with teleportation. This is all scaled by the end of this interesting video:
In The Flash Rebirth #3 (2009), Direct competition between the Flash and Superman did clearly demonstrate and acknowledge that during a chase where the Flash easily outruns Superman. See Thaddeus Howze's answer for the illustration.
And in Flashpoint (2011), the Flash does even better as he uses the Speed Force to travel back in time (some speed overflow you could say). Modern Age Superman can't time travel with his speed alone. See: Can Superman reverse time in comics by flying faster than speed of light?.
Everyone here doesn't realize Flash is faster then Superman by a lot no matter space or Earth. People are saying Flash can run at the speed of light and Superman can do it, then others are saying Superman flies faster in space. Regardless Flash is able to run so fast he can go back in time and he can travel through dimensions because he is so fast regardless. There have even been recent Flash comics where Mr and Ms Quick run so fast they can fly. Flash is the fastest man alive. Flash traveled at a speed faster than a Planck second which is the smallest period of time so basically Flash is able to beat time itself in a race.