A UK source reports that flat tyres account for about 10% of motor vehicle breakdowns.
A flat tyre can be repaired by a patch or plug; or the tyre may repair itself. Self-sealing tyres work on punctures up to a certain size.
The patch repair is commonly used in tyre shops. It may not be possible to patch a worn tyre if: the hole is close to a previous patch, there are already more than two patches, the puncture requires more than two patches, the punctures are too close, and/or the puncture is close to the sidewall. A patch is performed by removing the tyre, marking the puncture, scouring the surface to create a smooth surface (inside of the tyre), applying rubber cement, applying the patch, then pressing it into the surface with a small metal wheel attached to a handle. An alternative is a combination patch and plug. This is manufactured with a plug built into it; applying this patch is done similarly except with more steps, including drilling a hole at the puncture so the plug can be pulled through it, as well as cutting off the excess plug from the outside the tyre.
Tyres can leak air for a variety of reasons. These include, but are not limited to: damage to the wheel itself, a damaged valve stem, a puncture in the tyre (which can be hard to find if the puncturing object didn't embed itself in the tyre) and improper installation of the tyre, which could involve the bead of the tyre being cut when installed with excessive force.
Occasionally, a puncture may not "go all the way through" to the inside of the tyre. This may give the impression that a tyre is punctured but when the foreign object is removed the tyre may not lose pressure.
Also worth mentioning is the fact that tyres simply lose air over time. A brand new tyre, properly inflated, will lose air even with no punctures present. This is mainly due to the design of the valve stem, among other reasons. Given enough time, a tyre can fully deflate with no outside intervention.
Should one be so inclined, one way to locate the source of a leak in a tyre (a standard passenger tyre being used for this example) is to inflate the tyre to 50–60 psi (3.4–4.1 bar) and place the tyre and wheel in a tank of water with dish soap and watch for bubbles. This method may not work all the time, especially with very small punctures, but may help one find a possible leak.
The Image below shows the repairable zones. The green section is the area that can be reapired, the red sections shown are the area of the tyre that cannot be repaired.